# Full text of "Partial Differential Equations"

## See other formats

CO P3 OU 162031 >m OUP — 787— 1 3-6-75 — 1 (),()()(). OSMANIA UNIVERSITY LIBRARY Call No. £ 17 . 2>£" Accession No. Q Author /S 3>2- /^ Title This book sho'iitl b<* rriurncil <»n or before »hr date last marked belo PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS By H. BATEMAN This truly encyclopedic exposition of the methods of solving boundary value problems of mathematical physics by means of definite ana lytical expressions is valuable as both text and reference work. It covers an astonishingly broad field of prob- , lems and contains full references to classical and contemporary literature, as well as numerous examples on which the reader may test his skill. This edition contains a number of corrections and additional references furnished by the late Professor Bateman. The book includes sections on: Relation of the differential equations to varia tional principles, approximate solution of bound- ary value problems, method *of Ritz, orthogonal functions; classical equations, including uniform motion, Fourier series, free and forced vibrations, Heaviside's expansion, wave motion, potentials, Laplace's equation. Applic?^ns of the theorems of Green and Stokes; ' ><nann's method, elastic solids, fluid motion, torsion, membranes, electromagnetism; two dimensional problems, Fourier inversion, vi- bration of a loaded string and of a shaft; con- formal mapping, including the Riemann theorem, the distortion theorem, mapping of polygons. Equations in three variables, wave motion, teat flow; polar co-ordinates, Legendre polyno- nials, with applications; cylindrical co-ordinates, diffusion, vibration of a circular membrane; el- liptic and parabolic co-ordinates, with the cor- responding boundary problems; torodial co-ordi- nates and applications, ". . the book must be in the hands of every one who is interested in the boundary value problems of mathematical physics'*. — Bulletin of American Mathematical Society. Text in English. 6x9. xxii-f-522 pages, 29 illustrations. Originally published at $10.50. DOVER PUBLICATIONS 1780 Broadway, New York 19, N. Y. Please send me: G a copy of your new catalog D the following books: Name ¥ i PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS BY H. BAT EM AN, M.A., PH.D. Late Fellow of Trinity College, Cambridge ; Professor of Mathematics, Theoretical Physics and Aeronautics, California Institute of Technology, Pasadena, California NEW YORK DOVER PUBLICATIONS 1944 First Edition 1932 First American Edition 1944 By special arrangement with the Cambridge University Press and The Macmillan Co. Printed in the U. S. A. Dedicated to MY MOTHER CONTENTS PREFACE page xiii INTRODUCTION xv-xxii CHAPTER I THE CLASSICAL EQUATIONS §§ 1-11-1-14. Uniform motion, boundary conditions, problems, a passage to the limit. 1-7 §§ 1-15-1-19. Fourier's theorem, Fourier constants, Cesaro's method of summation, Parseval's theorem, Fourier series, the expansion of the integral of a bounded function which is continuous bit by bit. . 7-16 §§ 1-21-1-25. The bending of a beam, the Green's function, the equation of three moments, stability of a strut, end conditions, examples. 16-25 §§ 1*31-1-36. F^ee undamped vibrations, simple periodic motion, simultaneous linear equations, the Lagrangian equations of motion, normal vibrations, com- pound pendulum, quadratic forms, Hermit ian forms, examples. 25-40 §§ 1-41-1 -42. Forced oscillations, residual oscillation, examples. 40-44 § 1-43. Motion with a resistance proportional to the velocity, reduction to alge- braic equations. 44 d7 § 1-44. The equation of damped vibrations, instrumental records. 47-52 § 1-45-1 -46. The dissipation function, reciprocal relations. 52-54 §§ 1-47-1-49. Fundamental equations of electric circuit theory, Cauchy's method of solving a linear equation, Heaviside's expansion. 54-6Q §§ 1-51— 1-56. The simple wave-equation, wave propagation, associated equations, transmission of vibrations, vibration of a building, vibration of a string, torsional oscillations of a rod, plane waves of sound, waves in a canal, examples. 60-73 §§ 1-61-1 -63. Conjugate functions and systems of partial differential equations, the telegraphic equation, partial difference equations, simultaneous equations involving high derivatives, examplu. 73-77 §§ 1-71-1-72. Potentials and stream-functions, motion of a fluid, sources and vortices, two-dimensional stresses, geometrical properties of equipotentials and lines of force, method of inversion, examples. 77-90 §§ 1-81-1-82. The classical partial differential equations for Euclidean space, Laplace's equation, systems of partial differential equations of the first order fchich lead to the classical equations, elastic equilibrium, equations leading to the ^uations of wave-motion, 90-95 S 1*91. Primary solutions, Jacobi's theorem, examples. 95-100 $1'92./ The partial differential equation of the characteristics, bicharacteristics and rays. 101-105 ;§§ 1 '93-1 «94. Primary solutions of the second grade, primitive solutions of the wave-equation, primitive solutions of Laplace's equation. 105-111 §1-95. Fundamental solutions, examples. 111-114 viii Contents CHAPTER n APPLICATIONS OF THE INTEGRAL THEOREMS OF GREEN AND STOKES §§2*11-2-12. Green's theorem, Stokes' s theorem, curl of a vector, velocity potentials, equation of continuity. pages 116-118 §§ 2-13-2-16. The equation of the conduction of heat, diffusion, the drying of wood, the heating of a porous body by a warm fluid, Laplace's method, example. 118-125 §§ 2-21-2*22. Riemann's method, modified equation of diffusion, Green's func- tions, examples. 126-131 <§f 2-23-2*26. Green' s^theorem^for a general linear differential equation of the second order, characteristics, classification of partial differential equations of the second order, a property of equations of elliptic type, maxima and minima of solutions. 131-138 §§ 2-31-2-32. Green's theorem for Laplace's equation, Green's functions, reciprocal relations. ~ " 138-144 §§ 2-33-2-34. Partial difference equations, associated quadratic form, the limiting process, inequalities, properties of the limit function. 144-152 §§ 2-41-2-42. The derivation of physical equations from a variational principle, Du Bois-Reymond's lemma, a fundamental lemma, the general Eulerian rule, examples. 152-157 §§ 2-431-2-432. The transformation of physical equations, transformation of Eulerian equations, transformation of Laplace's equation, some special trans- formations, examples. 157-162 § 2-51. The equations for the equilibrium of an isotropic elastic solid. 162-164 § 2-52. The equations of motion of an inviscid fluid. 164-166 § 2-53. The equations of vortex motion and Liouville's equation. . 166-169 § 2-54. The equilibrium of a soap film, examples. 169-171 §§ 2-56-2-56. The torsion of a prism, rectilinear viscous flow, examples. 172-176 § 2-57. The vibration of a membrane. 176-177 §§ 2-58-2-59. The electromagnetic equations, the conservation of energy and momentum in an electromagnetic field, examples. 177-183 §§ 2-61-2-62. Kirchhoflfs formula, Poisson's formula, examples. 184-189 §§ 2-63-2-64. Helmholtz's formula, Volterra's method, examples. 189-192 §§ 2-71-2-72. Integral equations of electromagnetism, boundary conditions, the retarded potentials of electromagnetic theory, moving electric pole, moving electric and magnetic dipoles, example. 192-201 § 2-73. The reciprocal theorem of wireless telegraphy. 201-203 Contents IX CHAPTER IH TWO-DIMENSIONAL PROBLEMS § 3*11. Simple solutions and methods of generalisation of solutions, example, pages 204-207 §3*12. Fourier's inversion formula. 207-211 §§3-13-3-15. Method of summation, cooling of fins, use of simple solutions of a complex type, transmission of vibrations through a viscous fluid, fluctuating tem- peratures and their transmission through the atmosphere, examples. 211-215 § 3-16. Poisson's identity, examples. 215-218 § 3*17. Conduction of heat in a moving medium, examples. 218—221 § 3-18. Theory of the unloaded cable, roots of a transcendental equation, Kosh- liakov's theorem, effect of viscosity on sound waves in a narrow tube. 221-228 § 3-21. Vibration of a light string loaded at equal intervals, group velocity, electrical filter, torsional vibrations of a shaft, examples. 228-236 §§ 3-31-3-32. Potential function with assigned values on a circle, elementary treatment of Poisson's integral, examples. 236-242 §§ 3-33-3-34. Fourier series which are conjugate, Fatou's theorem, Abel's theorem for power series. 242-245 § 3-41. The analytical character of a regular logarithmic potential. • 245-246 § 3-42. Harnack's theorem. 246-247 § 3-51. Schwarz's alternating process. 247-249 § 3-61. Flow round a circular cylinder, examples. 249-254 § 3-71. Elliptic coordinates, induced charge density, Munk's theory of thin aerofoils. 254r~260 §§ 3-81-3-83. Bipolar co-ordinates, effect of a mound or ditch on the electric potential, example, the effect of a vertical wall on the electric potential. 260-265 CHAPTER IV CONFORMAL REPRESENTATION §§ 4-1 1-4-21. Properties of the mapping function, invariants, Riemann surfaces and winding points, examples. 266-270 §§ 4-22—4-24. The bilinear transformation, Poisson's formula and the mean value theorem, the conformal representation of a circle on a half plane, examples. 270-275 §§ 4-31-4-33. Riemann' s problem, properties of regions, types of curves, special and exceptional cases of the problem. 275-280 §§ 4-41-4-42. The mapping of a unit circle on itself, normalisation of the mapping problem, examples. 280-283 § 4-43. The derivative of a normalised mapping function, the distortion theorem and other inequalities. 283-285 § 4-44. The mapping of a doubly carpeted circle with one interior branch point. 285-287 §4-45. The selection theorem. 287-291 §4-46. Mapping of an open region. 291-292 b-2 x Contents § 4-51. The Green's function. pages 292-294 §§ 4-61-4-63. Schwarz's lemma, the mapping function for a polygon, mapping of a triangle, correction for a condenser, mapping of a rectangle, example. 294-305 § 4-64. Conformal mapping of the region outside a polygon, example. 305-309 §§ 4-71-4-73. Applications of conformal representation in hydrodynamics, the mapping of a wing profile on a nearly circular curve, aerofoil of small thickness. 309-316 §§ 4-81-4-82. Orthogonal polynomials connected with a given closed curve, the mapping of the region outside C', examples. 316-322 §§ 4-91-4-93. Approximation t*> the mapping function by means of polynomials, Daniell's orthogonal polynomials, Fe JOT'S theorem. 322-328 CHAPTER V EQUATIONS IN THREE VARIABLES §§ 5-11-5-12. Simple solutions and their generalisation, progressive waves, standing waves, example. 329-331 § 5-13. Reflection and refraction of electromagnetic waves, reflection and refrac- tion of plane waves of sound, absorption of sound, examples. 331-338 § 5-21. Some problems in the conduction of heat. 338-345 § 5-31. Two-dimensional motion of a viscous fluid, examples. 345-350 CHAPTER VI POLAR CO-ORDINATES §§ 6-11-6-13. The elementary solutions, cooling of a solid sphere. 351-354 §§ 6-21-6-29. Legendre functions, Hobson's theorem, potential function of degree zero, Hobson's formulae for Legendre functions, upper and lower bounds for the function Pn (/-i), expressions for the Legendre functions as nth derivatives, the associated Legendre functions, extensions of the formulae of Rodrigues and Conway, integral' relations, properties of the Legendre coefficients, examples. 354-366 §§ 6-31—6-36. Potential function with assigned values on a spherical surface $, derivation of Poisson's formula from Gauss's mean value theorem, some applica- tions of Gauss's mean value theorem, the expansion of a potential function in a series of spherical harmonics, Legendre's expansion, expansion of a polynomial in a series of surface harmonics. 367-375 §§ 6-41-6-44. Legendre functions and associated functions, definitions of Hobson and Barnes, expressions in terms of the hypergeometric function, relations be- tween the different functions, reciprocal relations, potential functions of degree n + \ where n is an integer, conical harmonics, Mehler's functions, examples. 375-384 §§ 6-51-6-54. Solutions of the wave-equation, Laplace's equation in n + 2 variables, extension of the idea of solid angle, diverging waves, Hankel's cylindrical functions, the method of Stieltjes, Jacobi's polynomial expansions, Wangerin's formulae, examples. 384-395 § 6-61. Definite integrals for the Legendre functions. 395-397 Contents xi CHAPTER VII CYLINDRICAL CO-ORDINATES § 7*11. The diffusion equation in two dimensions, diffusion from a cylindrical rod, examples. pages 398-399 § 7-12. Motion of an incompressible viscous fluid in an infinite right circular cylinder rotating about its axis, vibrations of a disc surrounded by viscous fluid, examples. 399-401 § 7« 13. Vibration of a circular membrane. 401 §7-21. The simple solutions of the wave-equation, properties of the Bessel functions, examples. 402-404 § 7-22. Potential of a linear distribution of sources, examples. 404-405 § 7-31. Laplace's expression for a potential function which is symmetrical about an axis and finite on the axis, special cases of Laplace's formula, extension of the formula, examples. 405-409 § 7-32. The use of definite integrals involving Bessei functions, Sommerf eld's expression for a fundamental wave-function, Hankel's inversion formula, examples. 409-412 § 7-33. Neumann's formula, Green's function for the space between two parallel planes, examples. * 412-415 § 7-41. Potential of a thin circular ring, examples. 416-417 § 7*42. The mean value of a potential function round a circle. 418-419 § 7-51. An equation which changes from the elliptic to the hyperbolic type. 419-420 CHAPTER VIII ELLIPSOIDAL CO-ORDINATES §§ 8-1 1-8-12. Confocal co-ordinates, special potentials, potential of a homogeneous solid ellipsoid, Maclaurin's theorem. 421-425 § 8-21 . Potential of a solid hypersphere whose density is a function of the distance from the 6entre. 426-427 §§ 8-31-8-34. Potential of a homoeoid and of an ellipsoidal conductor, potential of a homogeneous elliptic cylinder, elliptic co-ordinates, Mathieu functions, examples. 427-433 §§ 8-41-8-45. Prolate spheroid, thin rod, oblate spheroid, circular disc, con- ducting ellipsoidal column projecting above a flat conducting plane, point charge above a hemispherical boss, point charge in front of a plane conductor with a pit or projection facing the charge. 433-439 §§ 8-51-8-54. Laplace's equation in spheroidal co-ordinates, Lame products for spheroidal co-ordinates, expressions for the associated Legendre functions, spheroidal wave-functions, use of continued fractions and of integral equations, a relation between spheroidal harmonics of different types, potential of a disc, examples. 440-448 Xll Contents CHAPTER IX PARABOLOIDAL CO-ORDINATES § 9-11. Transformation of the wave-equation, Lam6 products. pages 449-451 §§ 0-21-0-22. Sonine's polynomials, recurrence relations, roots, orthogonal pro- perties, Hermite's polynomial, examples. 451-455 § 0-31. An erpression for the product of two Sonine polynomials, confluent hypergeometric functions, definite integrals, examples. 455-460 CHAPTER X TOROIDAL COORDINATES § 10-1 . Laplace's equation in toroidal co-ordinates, elementary solutions, examples. 461-462 § 10-2. Jacobi's transformation, expressions for the Legendre functions, examples. 463-465 § 10-3. Green's functions for the circular disc and spherical bowl. 465-468 § 10-4. Relation between toroidal and spheroidal co-ordinates. 468 § 10-5. Spherical lens, use of the method of images, stream-function. 468-472 §§ 10-6-10-7. The Green's function for a wedge, the Green's function for a semi- infinite plane. 472-474 § 10-8. Circular disc in any field of force. 474-475 CHAPTER XI DIFFRACTION PROBLEMS §§ 11-1-11-3. Diffraction by a half plane, solutions of the wave-equation, Som- merfeld's integrals, waves from a line source, Macdonald's solution, waves from a moving source. § 11-4. Discussion of Sommerf eld's solution. §§ 11 -5-1 1-7. Use of parabolic co-ordinates, elliptic co-ordinates. 476-483 483-486 486-490 CHAPTER XII NON-LINEAR EQUATIONS §12*1. Riccati's equation, motion of a resisting medium, fall of an aeroplane, bimolecular chemical reactions, lines of force of a moving electric pole, examples. 491-496 § 12-2. Treatment of non-linear equations by a method of successive approxi- mations, combination tones, solid friction. 497-501 § 12*3. The equation for a minimal surface, Plateau's problem, Schwarz's method, helicoid and catenoid. 501-509 § 12-4. The steady two-dimensional motion of a compressible fluid, examples. 509-511 APPENDIX LIST OF AUTHORS CITED INDEX 512-514 515-519 520-522 PREFACE TO AMERICAN EDITION The publishers are gratified that through the cooperation of The Macmillan Company and the Cambridge University Press, Professor Bateman's definitive work on partial differential equations is made available for text and reference use by American mathematicians and physicists in a reduced price, corrected edition. Professor Batemanthas kindly provided the corrections and addi- tions to references which are found on pages 11, 15, 24, 49, 50, 73, 124, 126, 127, 212, 219, 229, 230, 247, 261, 359, 364, 366, 403, 404, 410, 415, 417, 452, 454, 457, 462, 464, 477, 515, 517, 518 and 519. The correction required on page 219 was brought to Professor Bateman's attention by Mr. W. H. Jurney. DOVER PUBLICATIONS PREFACE IN this book the analysis has been developed chiefly with the aim of obtaining exact analytical expressions for the solution of the boundary problems of mathematical physics. In many cases, however, this is impracticable, and in recent years much attention has been devoted to methods of approximation. Since these are not described in the text with the fullness which they now deserve, a brief introduction has been written in which some of these methods are sketched and indications are given of portions of the text which will be particularly useful to a student who is preparing to use these methods. No discussion has been given of the partial differential equations which occur in the new quantum theory of radiation because these have been well treated in several recent book^, and an adequate discussion in a book of this type would have greatly increased its size. It is thought, however, that some of the analysis may prove useful to students of the new quantum theory. Some abbreviations jmd slight departures from the notation used in recent books have been adopted. Since the /,-notation for the generalised Laguerre polynomial has been used recently by different writers with slightly different meanings, the original T-notation of Sonine has been retained as in the author's Electrical find Optical Wave Motion. It is thought, however, that a standardised //-notation will eventually be adopted by most writers in honour of the work of Lagrange and Laguerre. The abbreviations '"eit" and "eif " used in the text might be used with advantage in the new quantum theory, together with some other abbrevia- tions, such as "oil" for eigenlrsrei and k'eiv" for eigenvector. The Heaviside Calculus and Ihe theory of integral equations are only briefly mentioned in the text: they belong rather to a separate subject which might be called the Integral Equations of Mathematical Physics. Accounts of the existence theorems of potential theory, Sturm-Liouville expansions and ellipsoidal harmonics have also been omitted. Many excellent books have however appeared recently in which these subjects are adequately treated. I feel indeed grateful to the Cambridge University Press for their very accurate work and intelligent assistance during the printing of this book. M. BATEMAN November 1931 INTRODUCTION THE differential equations of mathematical physics are now so numerous and varied in character that it is advisable to make a choice of equations when attempting a discussion. The equations considered in this book are, I believe, all included in some set of the form x « x ~ <> F — - 0 - 0 - - 0 to ' ¥.' "• to. ' where the quantities on the left-hand sides of these equations are the variational derivatives* of a quantity F9 which is a function of I independent variables xlt ... xlt of m dependent variables yl9 "... ym and of the deriva- tives up to order n of the t/'s with respect to the x's. The meaning of a variational derivative will be gradually explained. (T) The first property to be noted is that the variational derivatives of a function F all vanish identically when the function can be expressed in the form F_dol dGt dat JL — j r ~j r • • • i j j dx-i cLx^ axi where each of the functions G8 is a function of the x's and t/'s and of the derivatives up to order n — 1 of the i/'s with respect to the x'a. The notation d/dx8 is used here for a complete differentiation with respect to x8 when consideration is taken of the fact that F is not only an explicit function of xs but also an explicit function of quantities which are themselves functions of x8. Another statement of the property just mentioned is that the varia- tional derivatives of F vanish identically when the expression F dxldx2 ... dxl is an exact differential. In the case when there is only one independent variable x and only one dependent variable y, whose derivatives up to order n are respectively y'> y"y ••• y(n)9 the condition that Fdx may be an exact differential is readily found to be - dy dx \dy' dx*\dy" Now the quantity on the left-hand side of this equation is indeed the variational derivative of F with respect to y and will be denoted by the symbol SF/Sy. * For a systematic discussion of variational derivatives reference may be made to the papers of Th. de Donder in Bulletin de VAccuttmie BoyaU de Bdgique, Claase des Sciences (5), t. xv (1929-30). In some cases a set of equations must be supplemented by another to give all the equations in a set of the variational form. xvi Introduction In the case when F is of the form yN. (z) - zMx (y), where z is a function of x and Mx (y), Nx (z) are linear differential expres- sions involving derivatives up to order n and coefficients of these deriva- tives which are functions of x with a suitable number of continuous derivatives, we can say that the differential expressions Mx (y), Nx (z) are adjoint when 8F/8y ~ 0 for all forms of the function z. When z is chosen to be a solution of the differential equation Nx (z) = 0 the expression zM x (y) dx is an exact differential and so z is an integrating factor of the differential equation adjoint to Nx (z) = 0. The relation between two adjoint differential equations is, moreover, a reciprocal one, The idea of adjoint differential expressions was introduced by Lagrange and extended by Riemann to the case when there is more than one inde- pendent variable. Further extensions have been made by various writers for the case when there are several dependent variables*. Adjoint differential expressions and adjoint differential equations are now of great importance in mathematical analysis. A second important property of the variational derivatives may be introduced by first considering a simple integral and its first variation Integrating the (s-f l)th term s times by parts, making use of the equations it is readily seen that the portion of 87 which still remains under the sign of integration is It is readily understood now why the name " variational derivative " is used. The variational derivative is of fundamental importance in the Calculus of Variations because the Eulerian differential equation for a variational problem involving an integral of the above form is obtained by equating the variational derivative to zero. This rule is capable of extension, and rules for writing down the variational derivatives of a function F in the general case when there are * See for instance J. Kiirechak, Math. Ann. Bd. Lxn, S. 148 (1906); D. R. Davis, Trans. Amer. Math. Soc. vol. xxx, p. 710 (1928). Introduction xvii I independent variables and m dependent variables can be derived at once from the rules of § 2-42 for the derivation of the Eulerian equations. Since our differential equations are always associated with variational problems, direct methods of solving these problems are of great interest. The important method of approximation invented by Lord Rayleigh* and developed by W. Tlitzf is only briefly mentioned in the text, though it has been used by RitzJ, Timoshenko§ and many other writers|! to obtain approximate solutions of many important problems. An adequate dis- cussion by means of convergence theorems is rather long and difficult, and has been omitted from the text largely for this reason and partly because important modifications of the method have recently been suggested which lead more rapidly to the goal and furnish means of estimating the error of an approximation. In Ritz's method a boundary problem for a differential equation D (u) = 0 is replaced by a variation problem in which a certain integral / is to be made a minimum, the unknown function u being subject to certain supplementary conditions which are usually linear boundary conditions and conditions of continuity. The function ua, used by Ritz as an approxi- mation for u, is not generally a solution of the differential equation, but it does satisfy the boundary conditions for all values of the arbitrary con- stants which it contains. The result is that when an integral Ia is calculated from ua in the way that / is to be calculated from u, the integral Ia is greater than the minimum value Im of /, even when the arbitrary con- stants in ua are chosen so as to make Ia as small as possible. This means that Im is approached from above by integrals of the type Ia . Now it was pointed out by R. Courant ^[ that Im can often be approached from below by integrals Ib calculated from approximation functions ub which satisfy the differential equation but are subject to less restrictive supplementary conditions. If, for instance, u is required to be zero on the boundary, the boundary condition may be loosened by merely requiring ub to give a zero integral over the boundary in each of the cases in which it is first multiplied by a function vs belonging to a certain finite set. This idea has been developed by Trefftz** who uses the arithmetical mean of Ia and Ib * Phil. Trans. A, vol. CLXI, p. 77 (1870); Scientific Papers, vol. I, p. 57. t W. Ritz, Crette, Bd. cxxxv, S. 1 (1908); (Euvres, pp. 192-316 (Gauthier-Villars, Paris, 1911). J W. Ritz, Ann. der Phys. (4), Bd. xxvm, S. 737 (1909). § 8. Timoshcnko, Phil. Mag. (6), vol. XLVII, p. 1093 (1924); Proc. London Math. Soc. (2), vol. XX, p. 398 (1921); Trans. Amcr. Soc. Civil Engineers, vol. LXXXVII, p. 1247 (1924); Vibration Problems in Engineering (D. Van Nostrand, New York, 1928). || See especially M. Plancherel, Bull, dcs Sciences Math. t. XLVII, pp. 376, 397 (1923), t. XLVIII, pp. 12, 58, 93 (1924); Comptcs JRcndus, t. CLXIX, p. 1152 (1919); R. Courant, Acia Math. t. XLIX, p. 1 (1926); K. Friedrichs, Math. Ann. Bd. xcvni, S. 205 (1927-8). U R. Courant, Math. Ann. Bd. xcvn, S. 711 (1927). ** E. Trefftz, Int. Congress o£ Applied Mechanics, Zurich (1926), p. 135; Math. Ann. Bd. c, S. 603 (1928). xviii Introduction as a close approximation for Im , and uses the difference of Ia and Ib as an upper bound for the error in this method of approximation. This method is simplified by a choice of functions v8 which will make it possible to find simple solutions of the differential equation for the loosened boundary conditions. Sometimes it is not the boundary conditions but the conditions of continuity which should be loosened, and this makes it advisable not to lose interest in a simple solution of a differential equation because it does not satisfy the requirements of continuity suggested by physical con- ditions. In order that Ritz's method may be used we must have a sequence of functions which satisfy the boundary conditions and conditions of con- tinuity peculiar to the problem in hand. It is advantageous also if these functions can be chosen so that they form an orthogonal set, To explain what is meant by this we consider for simplicity the case of a single independent variable x. The functions 0! (#), ^2 (#)> •••> defined in an interval a < x < 6, are then said to form a normalised orthogonal set when the orthogonal relations = 1, ra= n are satisfied for each pair of functions of the set. This definition is readily extended to the case of several independent variables and functions defined in a domain R of these variables; the only difference is that the simple integrals are replaced by integrals over the domain of definition. The definition may be extended also to complex functions i/rn (x) of the form an (x) -f ifin (#)> where an (x) and /?n (x) are real. The orthogonal relations are then of type s f Ja (x) + ipm (x)] [an (x) - ipn (x)] dx=0, m*n, = 1, m = n. Many types of orthogonal functions are studied in this book. The trigonometrical functions sin (nx)9 cos (nx) with suitable factors form an orthogonal set for the interval (0, 2?r), the Legendre functions Pn (x), with suitable normalising factors, form an orthogonal set for the interval ( — 1, 1), while in Chapter ix sets of functions are obtained .which are orthogonal in an infinite interval. The functions of Laplace, which form the complete system of spherical harmonics considered in Chapter vi, give an orthogonal set of functions for the surface of a sphere of unit radius, and it is easy to construct functions which are orthogonal in the whole of space. In Chapter iv methods are explained by which sets of normalised orthogonal functions may be associated with a given curve or with a given area. In many cases functions suitable for use in Ritz's method of approximation Introduction xix are furnished by the Lam6 products defined in Chapters m-xi. These pro- ducts are important, then, for both the exact and the approximate solution of problems. It was shown by Ritz, moreover, that sometimes the functions occurring in the exact solution of one problem may be used in the ap- proximate solution of another; the functions giving the deflection of a clamped bar were in fact used in the form of products to represent the approximate deflection of a clamped rectangular plate. Early writers* using Ritz's method were content to indicate the degree of approximation obtainable by applying the method to problems which could be solved exactly and comparing the approximate solution with the exact solution. This plan is somewhat unsatisfactory because the examples chosen may happen to be particularly favourable ones. Attempts have, however, been made by Krylofff and others to estimate the error when an approximating function of order n, say Ua = $0 (%) + Cl<Al (X) + C2^2 (X)+ ..* + Cntn 0*0, is substituted in the integral to be minimised and the coefficients cs are chosen so as to make the resulting algebraic expression a minimum. Attempts have been made also to determine the order n needed to make the error less than a prescribed quantity e. ,- In Ritz's method a boundary problem for a given differential equation must first of all be replaced by a variation problerp. There are, however, modifications of Ritz's method in which this step is avoided. If, for in- stance, the differential equation is a variational equation 8F/8u = 0, the same set of equations for the determination of the constants cs is obtained by substituting the expression Ua for u directly in the equation tb Su . (SF/Su) dx = 0, J a and equating to zero the coefficients of the variations 8cs . This method has been recommended by HenckyJ and Goldsbrough§ ; it has the advantage of indicating a reason why in the limit the function ua should satisfy the differential equation. Another method, proposed by Boussinesq|| many years ago, has been called the method of least squares. If the differential equation is L9(u) = f(x), and a < x < b is the range in which it is to be satisfied with boundary * See, for instance, M. Paschoud, Sur V application de la mtihode de W. Ritz: These (Gauthier- Villars, Paris, 1914). f N. Kryloff, Comptes Rendus, t. CLXXX, p. 1316 (1925), t. CLXXXVI, p. 298 (1928); Annales de Toulouse (3), t. xix, p. 167 (1927). J H. Hencky, Zeits.fur angew. Math. u. Mech. Bd. TO, S. 80 (1927). § G. R. Goldsbrough, Phil. Mag. (7), vol. vn, p. 333 (1929). || J. Boussinesq, Ttieorie de la chaleur, 1. 1, p. 316. xx Introduction conditions at the ends, the constants c8 in an approximating function ua, which satisfies these boundary conditions, are chosen so as to make the integral [Lm(ua)-f(x)]*dx J a as small as possible. The accuracy of this method has been studied by Kryloff* who believes that Ritz's method and the method of least squares are quite comparable in usefulness. The method of least squares is, of course, closely allied to the well-known method of approximating to a function / (x) by a finite series of orthogonal functions the coefficients cs being chosen so that the integral")* rb If (*) - Ci<Ai 0*0 - C2</r2 (X) - ... ~ Cnifjn (X)]2 dx may be as small as possible. The conditions for a minimum lead to the equations & '.= f(s)j.(x)d8 («=l,2,...n). Ja For an account of such methods of approximation reference may be made to recent books by Dunham Jackson J, S. Bernstein§ and dc la Vallee Poussin||. In the discussion of the convergence of methods of approximation there is an inequality due to Bouniakovsky and Schwarz which is of fundamental importance. If the functions f (x), g (x) and the parameter c are all real, the integral f [/ (x) + ^ (#)]2 = A -f 2cH + c2B Ja is never negative and so AB — H2 > 0. This gives the inequality f [/ (x)Y dx 'b \jy (x)Y dx > f [/ (x) g (x)Y dx. J a * a J a There is a similar inequality for two complex functions f(x),g (x) and * N. Kryloff, Comptes Rend as, t. CLXI, p. 558 (1915); t. CLXXXI, p. 86 (11)25). Sec also Krawt- chouk, ibid. t. c'LXXxm, pp. 474, 992 (1926). t G. I'larr, Comptes Rcttilus, t. XLIV, p. 984 (1857); A* Tocpler, Arizctycr dcr Kais. Akad. zu Wu'n (1870), p. 205. | 1). Jackson, "The Theory of Approximation," Arner. Math. tioc. Colloquium publications, vol. xi (1930). § S. Hernstt'in, Lv^ona sur le# proprn'te* extremal?* rt In mcilleure approximation dcs fotictions unalytiqiu's <T une van able reelle (Gauthier-Villars, Paris, 1926). || C. J . do la Vallee Poussm, Lemons sur I' approximation des foncttons d'une variable reelle (ibid. 1919). Introduction xxi their conjugates / (x), g (x). Indeed, if c and c are conjugate complex quantities, the integral is never negative. Writing f=l+im, g=p+iq, c=£+iri, where I, m, p, q, £, 77 are all real, the integral may be written in the form A (£2 + r?2) + 2B£-f 2^ + D= ~[(A£ + £)*+ (A*i + < where A = (p2 + q2) dx = gg dx, J a Ja D = f (i2 + m2) dx = ( //da;, ' Ja J a B + iC = f fgdx, B - iC = [ Ja Ja Since the integral is never negative, we have the inequality AD > B* + C2, which may be written in the form* In this inequality the functions / and g may be regarded as arbitrary integrable functions. This inequality and the analogous inequality for finite sums are used in § 4-81. In the approximate treatment of problems in vibration the natural frequencies are often computed with the aid of isoperimetric variation problems. Ritz's method is now particularly useful. If, for instance, the differential equation is and the end conditions u (a) = 0, u (b) = 0, the aim is to make the integral a minimum when the integral f [«(*)]• «fe Ja * This inequality is called Schwarz's inequality by E. Schmidt, Rend. Palermo, t. xxv, p. 58 (1908). xxii Introduction has an assigned value. This is accomplished by replacing u by a finite series in both integrals and reducing the problem to an algebraic problem. It was noted by Rayleigh that very often a single term in the series will give a good approximation to the frequency of the fundamental frequency of vibration. To obtain approximate values of the frequencies of overtones it is necessary, however, to use a series of several terms and then the work becomes laborious as it is necessary to solve an algebraic equation of high order. Many other methods of approximating to the frequencies of over- tones are now available. Trefftz has recently introduced a new method of approximating to the solution of a differential equation, in which the original variation problem 87 = 0 is replaced by a modified variation problem 87 (<r) = 0 in such a way that the desired solution u can be expressed in the form This method, combinfed with Trefftz's method of estimating the error of approximation to an integral such as / (e), can lead to an estimate of the error involved in a computation of u. In the problem of the deflection of a clamped plate under a given distribution of load, the function / (e) represents the potential energy when a concentrated load € is placed at the point where the deflection u is required. Courant has shown that the rapidity of convergence of a method of approximation can often be improved by modifying the variational problem, introducing higher derivatives in such a way that the Eulerian equatioA of the problem is satisfied whenever the original differential equation is satisfied. This device is useful also in applications of Trefftz's method. An entirely different method of approximation is based on the use of difference equations which in the limit reduce to the differential equation of a problem. The early writers were content to adopt the principle, usually called Rayleigh's principle, that it is immaterial whether the limiting pro- cess is applied to the difference equations or their solutions. Some attempts have been made recently to justify this principle* and also to justify the use of a similar principle in the treatment of problems of the Calculus of Variations by a direct method, due to Euler, in which an integral is replaced by a finite sum. An example indicating the use of partial difference equations and finite sums is discussed in § 2-33. * See the paper by N. Bogoliouboff and N. Kryloff, Annals of Math. vol. xxrx, p. 255 (1928). Many references to the literature are contained in this paper. In particular the method is discussed by R. B. Robbing, Amer. Journ. vol. xxxvn, p. 367 (1915). CHAPTER I THE CLASSICAL EQUATIONS § 1-11. Uniform motion. It seems natural to commence a study of the differential equations of mathematical physics with a discussion of the equation ^ 3*"°' which is the equation governing the motion of a particle which moves along a straight line with uniform velocity. It may be thought at first that this equation needs no discussion because the general solution is simply x = At + B, where A and B are arbitrary constants, but in mathematical physics a differential equation is almost always associated with certain supple- mentary conditions, and it is this association which presents the most interesting problems. A similar differential equation describes an essential property of a straight line, when x and y are inter- preted as rectangular co-ordinates, and its solution y = mx + c is the f amiliar equation of a straight line : the property in question is that the line has a constant direction, the direction or slope of the line being specified by the constant m. For some purposes it is convenient to regard the line as -a ray of light, especially as the conditions for the reflection and refraction of rays of light introduce interesting supplementary or boundary conditions, and there is the associated problem of geometrical foci of a system of lenses or reflecting surfaces. If a ray starts from a point Q on the axis of the system and is reflected or refracted at the different surfaces of the optical system it will, after completely traversing the system, be transformed into a second ray which meets the axis of the system in a point Q, which is called the geometrical focus of Q. The problem is to find the condition that a given point Q may be the geometrical focus of another given point Q. This problem is generally treated by an approximate < method which illustrates very clearly the mathematical advantages gained by means of simplifying assumptions. It is assumed that the angle between the ray and the axis is at all times small, so that it can be represented at any time by dy/dx. 2 The Classical Equations Let y2 '= 4a# give an approximate representation of a refracting surface in the immediate neighbourhood of the point (0, 0) on the axis. If y is a small quantity of the first order the value of x given by this equation can be regarded as a small quantity of the second order if a is of order unity. Neglecting quantities of the second order we may regard x as zero and may denote the slope of the normal at (#, y) by _ dx „ y dy 2a' Now let suffixes 1 and 2 refer to quantities relating to the two sides of the refracting surface. Since the angle between a ray and the normal to the refracting surface is approximately dy/dx -f- y/2a, the law of refraction is represented by the equations Denoting by [u] the discontinuity u2 — u± in a quantity u, we have the boundary conditions [dy\ [y] = o. Dropping the suffixes we see that these boundary conditions are of type [y] = o, where A and B are constants which may be either positive or negative. In the case of a moving particle, which for the moment we shall regard as a billiard ball, a supplementary condition is needed when the ball strikes another ball, which for simplicity is supposed to be moving along the same line. If Ui , u2 are the velocities of the first ball before and after collision, U19 U2 those of the second ball before and after collision, the laws of impact give u>- U,= - e (u, - 17,), mu2 + MU2 = where e is the coefficient of restitution and m, M are the masses of the two balls. Regarding Ul as known and eliminating U2 we have 0 (M -f- m) u2 = (m - eM) u± + M (1 + e) Ul . Replacing u2 — u± by [dx/dt] we have the boundary conditions for the collision [x] =0, ( M 4- m) [g] = M ( 1 + e) ( U, - Boundary Conditions 3 These hold for .the place x = xl where the collision occurs, x being the co- ordinate of the centre of the colliding ball. The boundary conditions considered so far may be included in the general conditions , ' [y] = o, where A, JB and C are constants associated with the particular boundary under consideration. § 1-12. Other types of boundary condition occur in the theory of uni- form fields of force. A field of force is said to be uniform when the vector E which specifies the field strength is the same in magnitude and direction for each point of a certain domain D. Taking the direction to be that of the axis of x the field strength E may be derived from a potential V of type V — Ex by means of the equation ,„ E = - , > ax V being an arbitrary constant. This potential V satisfies the differential equation /2 r/ throughout the domain D. Boundary conditions of various types are suggested by physical con- siderations. At the surface of a conductor V may have an assigned value. At a charged surface -j- may have an assigned value, while there may be a surface at which [F] has an assigned value (contact difference of potential). With boundary conditions of the types that have already been con- sidered many interesting problems may be formulated. We shall consider only two. § 1-13. Problem 1. To find a solution of d2y/dx2 = 0 which satisfies the conditions y = 0 when x = 0 and when x = 1 ; [dy/dx] — — 1 when x — £ ; [y] = o- The first condition is satisfied by writing y = Ax x< g = JS(l-a;) x>£. The condition [y] = 0 gives A£ = B (1 - fl, 4 The Classical Equations and the condition [dy/dx] = — 1 gives A + B=\. .: A=l-f, B = f Hence y = 9 (x, £) = a; (1 - £) (a; < This function is called the Green's function for the differential expression d2y/dx2, on account of its analogy to a function used by George Green in the theory of electrostatics. It may be remarked in the first place that a solution of type P + Qx which satisfies the conditions y ~ a when x = 0, y = b when x = 1, is given by the formula Secondly, it will be noticed that </ (a;, £) is a symmetrical function of x and £; in other words g (x, £) = g (£, #)• A third property is obtained by considering a solution of d2y/dx2 — 0 which is a linear combination of a number of such Green's functions, for example, n y= £ fsg (x, £,), s-l where flt /2, /3, ... are arbitrary constants. The derivative dy/dx drops by an amount/j at gl9 by an amount /2 at £2, and so on. * Let us now see what happens when we increase the number of points f i , f 2 > & > • • • and proceed to a limit so that the sum is replaced by an integral y=flog(x,$)f(€)d£ ...... (A) JO We find on differentiating that = - xf(x) - (1 - x)f(x) = -/(«), the function / (x) being supposed to be continuous in the interval (0, 1). It thus appears that the integral is no longer a solution of the differential equation d^/dx2 = 0, but is a solution of the non-homogeneous equation Conversely, if the function / (x) is continuous in the interval (0, 1) a solution of this differential equation and the boundary conditions, y = 0 when x = 0 and when x = 1, is given by the formula (A); this formula, The Green's Function 5 moreover, represents a function which is continuous in the interval and has continuous first and second derivatives in the interval. Such a function will be said to be continuous (D, 2), or of class C" (Bolza's notation). § 1-14. Problem 2. To find a solution of d2y/dx2 = 0 and the supple- mentary conditions [y]=° U*-,M y = 0 when x = 0 and when x = 1 ; n [dyjdx] + k*y = 0) ' ' where s = 1, 2, 3, ... n — 1. Let y = ^48a; -f #8> 5 — I < nx < s, then the supplementary conditions give B± = 0, ^4n -f J5n — 0, n ?! = 0, 7i (A2 - A,) -f £2 (AJn -f ^) = 0, (A3 - A,) | + £3 - J?2 = 0, n (A3 - AJ + k* (2A2/n + B2) = 0, H nB3 = (^! + ^42 -f ... A3) - sA8, n2 (AM - A8) + P (Al + A2+ ... ^f) = 0, ** (A*+i - 2^« + A*-i) + &AB =0, s > 1. This difference equation may be solved by writing k2 = 2n2 (1 — cos 6), .-. ^8 = A! cos (« - 1) 0 + K sin (s - 1) 0, where K is a constant to be determined. Now A2 = Ai + 2Al (cos 6 - 1) = Al (2 cos 0 - 1), therefore K sin 0 = Al (cos 0 — 1), _ . A sin sQ — sin 5— and so As = Al - The condition 0 = An -f Bn is satisfied if nO = r-n, where r is an integer. In the limit when n = oo this condition becomes k = lim 2n sin — = TTT, 6 The Classical Equations and this is exactly the condition which must b'e satisfied in order that the differential equation may possess a non- trivial solution which satisfies the boundary conditions y = 0 when x = 0 and when x = 1. The general solution of this equation is, in fact, ~ , r\ • i y — " cos kx -f Q sin to, where P and $ are arbitrary constants. To make y = 0 when # = 0 we choose P = 0. The condition y = 0 when x = 1 is then satisfied with Q =£ 0 only if sin & ^ 0, i.e. if k = r?r. The exceptional values of P of type (rTr)2 are called by the Germans " Eigenwerte " of the differential equation (B) and the prescribed boundary conditions. A non-trivial solution Q sin (kx) which satisfies the boundary conditions is called an "Eigenfunktion." These words are now being used in the English language and will be needed frequently in this book. To save printing we shall make use of the abbreviation eit for Eigenwert and eif for Eigenfunktion. The conventional English equivalent for Eigenwert is characteristic or. proper value and for Eigenfunktion proper function. The theorem which has just been discussed tells us that the differential equation (B) and the prescribed boundary conditions have an infinite number of real eits which are all simple inasmuch as there is only one type of eif for each eit. The eits are, moreover, all positive. The quantities &r2 = ( 2n sin -- j may be regarded as eits of the differential equation d2y/dx* = 0 and the preceding set of boundary conditions. These eits are also positive, and in the limit n ->oo they tend towards eits of the differential equation (B) and the associated boundary conditions. The solution y corresponding to k is, for s — 1 < nx < 5, TT\ [f S\ ( . STTT . S - 1 . nrX 1 . sr-rrl )\(x — -sin ---- sin -- -f - sin — , ...... (C) nl [\ n> \ n n J n n \ v ' and it is interesting to study the behaviour of this function as n -> oo to see if the function tends to the limit •*i-i . f ^si TTT T , ., A A fT7r\ Let us write A0 = A1(-~} cosec 0 1\nJ \n A A ^o (x) = r7r° sin (rirx), Fl (x) = — x sin (mx), and let us use F (x) to denote the function (C) which represents a potygon with straight sides inscribed in the curve y — FQ (x). A Passage to the Limit 7 The closeness of the approximation of F (x) to F1 (x) can be inferred from the uniform continuity of FQ (x). Given any small quantity 6 we can find a number n (e) such that for any number n greater than n (c) we have the inequality r. (*)-*•('- < € < for any point x in the interval s — 1 < nx < s and for any value of s in the set 1, 2, 3, ... n. In particular - 1\ _ /s\ n ) ° \n) Now F (x) = FQ (-} + (s- nx) \FQ (S ~ l] - F0 (-][ (s- Knx< s). \n/ [_ \ n / \n/ j Therefore | F (x) - FQ (x) \ < e + c. On the other hand IV (/y\ V I fy\ I I V { rr\ •^0 \x) ~ * I \X) I ~ I ^0 \X) Therefore < A1\-- cosec 1 IV | ^ (a;) - ^ (a;) | < 2e + ^ cosec ~~ - But when € is given we can also choose a number m (e) such that for n > m (e) we have the inequality A [TTT TTT _ "1 I , — cosec ----- 1 l[_n n J I < €. Consequently, by choosing n greater than the greater of the two quantities n (e) and m (e), if they are not equal, we shall have \F(x)~ F, (x) | < 36. This inequality shows that as n -> oo, F (x) tends uniformly to the limit F, (x). This method of obtaining a solution of the equation 2 +*•»-« from a solution of the simpler equation d2y/dx2 = 0 by a limiting process, can be extended so as to' give solutions of other differential equations and specified boundary conditions, but the question of convergence must always be carefully considered. § 1*15. Fourier's theorem. It seems very nat.ural to try to find a solution of the equation and a prescribed set of supplementary conditions by expanding / (x) in a 8 The Classical Equations d?ii series of solutions of -—£ + k*y = 0 and the prescribed supplementary con- ditions, because if f(x)=Xbnsinnx (A) the differential equation is formally satisfied by the series ~ , sin nx y.S&.—i-, and if the original series is uniformly convergent the two differentiations term by term of the last series can be justified. When the f unction /(#) is continuous it is not necessary to postulate uniform convergence because Lusin has proved that if the series (A) converge at all points of an interval / to the values of a continuous function/ (x) then the series (A) is integrable term by term in the interval 7. Unfortunately it has not been proved that an arbitrary continuous function can be expanded in a trigonometrical series. Indeed, we are faced with the question of the possibility of expanding a given function / (x) in a trigonometrical series of type (A). This question is usually made more definite by stipulating the range of values of x for which the representation of / (x) is required and the type of function/ (x) to which the discussion will be limited. A mathematician who starts out to find an expansion theorem for a perfectly arbitrary function will find after mature consideration that the programme is too ambitious*, as there are functions with very peculiar properties which make trouble for the mathematician who seeks complete generality. It is astonishing, however, that a function represented by a trigonometrical series is not of an exceed- ingly restricted type but has a wide degree of generality, and after the discussions of the subject by the great mathematicians of the eighteenth century it came as a great surprise when Fourier pointed out that a trigonometrical series could represent a function with a discontinuous derivative, and even a discontinuous function if a certain convention were adopted with regard to the value at a point of discontinuity. In Fourier's work the coefficients were derived by a certain rule now called Fourier's rule, though indications of it are to be found in the writings of Clairaut, Euler and d'Alembert. In the case of the sine-series the rule is that 2 [w bn = - / (x) sin nxdx, it Jo and the range in which the representation is required is that of the interval (0 < x < TT). When the range is (0 < x < 2ir) and the complete trigono- metrical series * « / (x) = $0o + 2 an cos nx n-l 00 + S bnsinnx n-l * For the history of the subject see Hobson's Theory of Functions of a Real Variable and Burk- hardt's Report, Jahresbericht der Deuteehen Math. Verein, vol. x (1908). Fourier Constants 9 is to be used for the representation, Fourier's rule takes the form an = - / (x) cos nxdx, TT JQ 1 f2tr bn = - / (x) sin nx dx, (B) TT Jo and the coefficients an , bn are called the Fourier constants of the function Unless otherwise stated the symbol/ (x) will be used to denote a function which is single-valued and bounded in the interval (0, 2n) and defined out- side this interval by the equation / (x -f 277) = / (x). For some purposes it is more convenient to use the range (— TT < u < TT) and the variable u = 2?r — x. If / (x) = F (u) the coefficients in the ex- pansion of F (u) in a trigonometrical series of Fourier's type are given by formulae exactly analogous to (B) except that the limits are — TT and TT instead of 0 and 2?r. The advantage of using the interval (— TT, TT) instead of the interval (0, 277-) is that if F (u) is an odd function of u, i.e. if F (— u) = — F (u), the coefficients an are all zero, and if F (u) is an even function of u, i.e. if F (— u) — F (u), the coefficients bn are all zero. In one case the series becomes a sine-series and in the other case a cosine-series. The possibility of the expansion of / (x) in a Fourier series is usually established for a function of limited variation*, that is a function such that the sum n_1 s I /(*.«)-/(*.) I s~0 is bounded and < N9 say, for all sets of points of subdivision xl9x2y ... a?n-1 dividing the interval (0, 2n) up into n parts and for all finite integral values of n. Such a function is also called a function of limited total fluctuation and a function of bounded variation. In addition to this restriction on / (x) it is also supposed that the integrals in the expressions for the coefficients exist in the ordinary sensef. In the case when the integral representing an is an improper integral it is assumed that the integral 2 (C) is convergent. If x is any interior point of the interval (0, 27r) it can be shown that when the foregoing conditions are satisfied the series is con- vergent and its sum is c Mm * Whittaker and Watson's Modem Analysis, 3rd ed. p. 175. f That is, in the Riemann sense. There are corresponding theorems for the cases in which other definitions of integral (such as those of Stieltjes and Lebesgue) are used. 10 The Classical Equations when the limits oif(x± e) exist, i.e. with a convenient notation H/(* + 0) +/(s - 0)] =/(x), say. When the function / (x) is continuous in an interval (a < x < /?) con- tained in the interval (0, 2n), is of limited variation in the last interval and the other conditions relating to the coefficients are satisfied, it can be shown * that the series is uniformly convergent for all values of x for which « -f S < # -' /J — 8, where S is any positive number independent of x. When the conditions of continuity and limited variation are dropped and the function / (x) is subject only to the conditions relating to the existence of the integrals in the formulae for the coefficients and the convergence of the integral (C), there is a theorem due to Fejcr, which states thatf />) = lim i {A0 + S, (x) + S2 (x) + ... S,K_, (x)}, m*-± oo •ft' in where A0 = |a0, An (x) = an cos nx + bn sin nx, Sm(x)= 2 An (x). «-<) This means that the series is summable in the Cesaro sense by the simple method of averaging which is usually denoted by the symbol (C, 1). This is a theorem of great generality which can be used in applied mathematics in place of Fourier's theorem. It is assumed, of course, that the limits lim /(a?+€)=/(a? + 0), Km /(*?-€)=/(*- 0) t-> 0 «-> 0 exist J. § 1-16. Cesaro's metJwd of summation §. Let n8n = sl + 82+ ... + sn, then, if Sn -> S as n -> oo, the infinite series (1) is said to be summable (C, 1) with a Cesaro sum S. For consistency of the definition of a sum it must be shown that when the series (1) converges to a sum s, we have 8 = S. To do this we choose a positive integer n, such that I Sn+p-*n | <.€ ...... (2) for all positive integral values of p. This is certainly possible when s exists and we have in the limit |*-*n| <*. ...... (3) * Whittaker and Watson, Modern Analysis, 3rd ed. p. 179. t Ibid. p. 169. J When/ (a: -f 0) = / (x - 0) this implies that f(x) is continuous at the point x. § Bull, des Sciences Math. (2), t, xrsr, p. 114 (1890). See also Bromwich's Infinite Series. Fejer's Theorem 11 Now let v be an integer greater than n and let Cm be defined by the equation ~ vCm+1 = v-m, ...... (4) then Sv = c^Ui 4- c2^2 4- ... 4- cvuv. ...... (5) But Cj > c2 > ... > cv > 0, hence it follows from (2) that I cn+l^n+l 4- Cn-f2^n+2 ~H ••• 4" CVUV \ < €Cn4.1, i-e. I S, - (q^ 4- C2u2 -f ... 4- cn^n) | < ecn+1. Making v -> oo we see that if S be any limit of $„ |S-a»| <€. ...... (6) Combining (3) and (6) we find that | S - s | < 2e. Since e is an arbitrary small positive quantity it follows that S = s and so the sequence $„ has only one limit s. § 1-17. Fejer's theorem. Let us now write u^ = ^40, ?/n+1 = ^4n (x), then, by using the expressions for the cosines as sums of exponentials, it is readily found that* , where 26 = | a; — ^ | . Now the integrand is a periodic function of / of jjb^ioil 277, Consequently we may also write b^arthermore, since «lr'^ = /w 4. 2 (w - 1) cos 26 + 2 (m - 2) cos 46 + . . . + 2 cos 2 (m - 1) 9 sm'20 •i • i-i xu A f|7r sin2 w0 | it fs readily seen that ---.- -9 « = ^TT. J Jo msm2^ I Writing ^ (0) = / (x 4- 20) + / (* - 20) - 2/ (a;), and inaking use of the last equation, we find that w ier« <k (0) -> 0 as 0 -> '0. How if e is any small positive quantity we can choose a number 8 The details of the analysis are given in Whittaker and Watson's Modern Analysis 12 The Classical Equations whenever 0 < 9 < 8, and if € is independent of m the number 8 may be regarded as independent of m. Writing for brevity sin2 m6 = m sin2 9P (0), TT - 2« and noting that P (6) is never negative, we have o (9) fa (9) d9 P P (9) | +x (9) | d9 + p P (9) \ <t>x (9) \ d9 Jo Js < Let us now suppose that | / (t) \ dt exists, then J — TT f I & (9) I d9 Jo also exists, and by choosing a sufficiently large value of m we can make aem sin2 8 > f" I J> (9) I d9. Jo This makes the second integral on the right of (B) less than «e, which is also the value of the first integral. Therefore \Sm(x)-f(x)\ <2ae/7r=e; consequently Sm (x) -> / (x) as m -> oo. When/ (x) is continuous throughout the interval (— TT < x < TT) all! ,i foregoing requirements are satisfied and in addition / (x) = f (x) ; col & quently, in this case, Sn (x) -> f (x), and this is true for each point I o the interval. \ f This celebrated theorem was discovered by Fejer*. The conditional oJ the theorem are certainly satisfied when the range (— TT < x < TT) canpbe divided up into a finite number of parts in each of which / (x) is bo <mded and continuous. Such a function is said to be continuous bit by bit (Stiickweise stetig); the Cesaro sum for the Fourier series is then/ (x) at any point of the range, / (x) and / (x) being the same except at the points of subdivision. § 1-18. ParsevaTs theorem. Let the function / (x) be continuous bit by bit in the interval (— TT, ?r) and let its Fourier constants be an, 6n; it then be shown that [/ (*)? dx = TT [~K2 + 2 (an2 -f 6n2)l ....... (A) L n=l J Math. Ann. Bd. LVIII, S. 51 (1904). ParsevaVs Theorem 13 We shall find it convenient to sum the series (A) by the Ces&ro method* This will give the correct value for the sum because the inequality ;-f n-l n-1 indicates that the series is convergent. To find the sum (C, 1) we have to find the limit of Sm where, by a simple extension of 1-17 (A), S -- M ~ 277 _ .^ sn* 29 being equal to | x — t \ . Since the region of integration can be divided up into a finite number of parts in each of which the integrand is a continuous function of x and t, the double integral exists and can be transformed into a repeated integral in which x and 0 are the new independent variables. The region for which 6 lies between 6Q and 00 -f dd, while x lies between XQ and XQ -h dx consists of two equal partsf; sometimes two, sometimes one and sometimes none of these parts lie within the region of integration. When this is taken into consideration the correct formula for the transformation of the integral is found to be 1 fir .= H- *TTJQ n-20 20) fTr JW-T In the derivation of this result Fig. 1 will be found to be helpful. The lines MlM2y MZM± are those on which 0 has an assigned value, while N1N2, JV3A74 are lines on which 0 has a different assigned value. It will be noticed that a line parallel to the axis of t meets M1M2 , M3M^ either once or twice, while it meets N1N29 N^N^ either once or not at all. Applying the theorem of § 1-17 to (B) we get -7T ...... (B) 7T-20 7T Fig. 1. - r f(x)/(x)dx, J — IT and when/ (x) is defined to be/ (x) this result gives (A). lim , m->oo * This is the plan adopted in Whittaker and Watson's Modern Analysis, p. 181. The present proof, however, differs from that given in Modern Analysis, which is for the case in which f (x) is bounded and integrable. t It will be noted that the Jacobian of the transformation has a modulus equal to two. 14 The Classical Equations The theorem (A) was first proved by Liapounoff*; the present investi- gation is a modification of that given by Hurwitzf. Now let F (x) be a second function which is continuous bit by bit in the interval — -n < x < 77 and let An , Bn be its Fourier constants. Applying the foregoing theorem to F (x) -f / (x) and F (x) — f (x), we obtain T [F (x) + f (x)Y dx = 77 [i (A0 + a0)2 -f S {(4n + aj2 -f J-7T L W=l I" [F (x) - f (x)]* dx - TT [4 Mo - a0)2 + S {(4n - a J2 + ./-* L n-1 / Subtracting, we obtain the important formula I* / (x) F (x) dx = >n \!>A0a0 + 2 (,4nan + Bnbn) J-1T L W-l which is usually called Parseval's theorem, though ParsevaFs derivation of the formula was to some extent unsatisfactory. In the modern theory, when Lebesgue integrals are used, the theorem is usually established for the case in which the functions f (x), F (x), [f (x)]2 and [F (x)]2 are integrable in the sense of Lebesgue. There is also a converse theorem which states that when the series (A) converges there is a function / (x) with an and 6n as Fourier constants which is such that [/ (#)]2 is integrable and equal to the sum of the series. This theorem was first proved by Riesz and Fischer. Several proofs of the theorem are given in a paper by W. H. Young and Grace Chisholm Young J. The theorem has also been extended by W. H. Young §, the complete theorem being also an?> extension of Parseval's theorem. A general form of Parseval's theorem has been used to justify the integration term by term of the product of a function and a Fourier series. ADDITIONAL RESULTS 1. If the functions / (x), F (x) are integrable in the sense of Lebesgue, and [/(z)]2, [F (x)]2 are also integrable in the same sense, then|| T W J -T x)F(t)dt = K^o+ 2 (anAn H- bnBn) cos nx - 2 (anBn - bnAn) sin nx. 2. If / (x) is a periodic function of period 2?r which is integrable in the sense of Lebesgue, and if g (x) is a function of bounded variation which is such that the integral '00 \g(x)\ dx o * Comptes Rendus, t. cxxvi, p. 1024 (1898). t Math. Ann. Bd. LVII, S. 429 (1903). J Quarterly Journal, vol. XLJVI p. 49 (1913). ^ Comptes Rendus, t. CLV, pp. 30, 472 (1912); Proc. Roy. Soc. London, A, vol. LXXXVII, p. 331 (1912); Proc. London Math. Soc. (2), vol. xii, p. 71 (1912). See also F. Hausdorff, Math. Zeits. Bd. xvi, S. 163(1923). || W. H. Young, Comptes Rendus, t. CLV, p. 30 (1912); Proc. Roy. Soc. London, A, vol. LXXXVII, p. 331 (1912). Fourier Series 15 is convergent, then the value of the integral r/(x)g(x)dx Jo may be calculated by replacing / (x) by its Fourier series and integrating formally term by term. In particular, the theorem is true for a positive function g (x) which decreases steadily as x increases and is such that the first integral is convergent. [W. H. Young, Proc. London Math. Soc. (2), vol. ix, pp. 449, 463 (1910); vol. xin, p. 109 (1913); Proc. Roy^oc. A, vol. xxxv, p. 14 (1911). G. H. Hardy, Mess, of Math. vol. LI, p. 186(1922).] § 1-19. The expansion of the integral of a bounded function which is continuous bit by bit. If in Parse val's theorem we put F (x) = 1 , - TT < x < z, F (x) = 0, z < x < TT, we have A^ = - F (x) dx = - dx = - n , TT J -n 7T J-TT TT An = - cos nx.dx = [sin nz], TT ) -n n-n Bn = - sin nx.dx — — [cos nn — cos nz], TT j _ff nrr and we have the result that JZ co ] / (x) dx = |a0 (z -f 77) -|- 2 - [an sin nz -\- bn (cos WTT — cos nz)]. -TT n=-i n ...... (A) Now the function ^a0z can be expanded in the Fourier series - 1 — «o 2 ~ cos mr sin /i^, «= i hence the integral, on the left of (A) can be expanded in a convergent trigonometrical series. To show that this is the Fourier series of the function we must calculate the Fourier constants. n dx . . an a() H — — / (2) c^s nz = ----- cos mr, J v ' n n 1 [n [z 1 (n Now - sin nz dz f (x) dx = ---- cos mr f (x) 7Tj-n J—rr M™ J -n If71" dz p . . — / (2) TT ]-„ n J v ' 1 f* [s 1 [" cos nzdz f (x) dx = — dz sin nz f (z) TTJ-rr J-n HIT J -n -^- - f dz f / (x) dx=\* f (x) dx - ! f zf(z) dz TT)-* J-n J -n ^ J -n OJ J TTO,, + 2 - &„ cos n?r, 16 The Classical Equations by Parseval's theorem. Hence the coefficients are precisely the Fourier constants and so the integral of a function which is continuous bit by bit can be expanded in a Fourier series. This means that a continuous periodic function with a derivative continuous bit by bit can be expanded in a Fourier series. Proofs of this theorem differing from that in the text are given by Hilbert-Courant, Methoden der Mathematischen Physik, Bd. I (1924), and by M. G. Carman, Bull. Amer. Math. Soc. vol. xxx, p. 410 (1924). It should be noticed that equation (A) shows that when / (x) can be expanded in a Fourier series this series can be integrated term by term. A more general theorem of this type is proved by E. W. Hobson, Journ. London Math. Soc. vol. n, p. 164 (1927). Fourier's theorem may be extended to functions which become infinite in certain ways in the interval (0, 277). When the number of singularities is limited the singularities may be removed one by one by subtracting from / (x) a simple function hs (x) with a singularity of the same type. This process is continued until we arrive at a function 0 (*)=/(*)- 2 h.(x) S-l which does not become infinite in the interval (0, 277). The problem then reduces to the discussion of the Fourier series associated with each of the functions hs (x). § 1-21. The bending of a beam. We shall now consider some boundary problems for the differential equation d*yfdx* = 0, which is the natural one to consider after d*y/dx2 = 0 from the historical standpoint and on account of the variety of boundary conditions suggested by mechanical problems. The quantity y will be regarded here as the deflection from the equi- librium position of the central axis of a long beam at a point Q whose distance from one end is x. The beam will be assumed to have the same cross-section at all points of its length and to be of uniform material, also the deflection at each point will be regarded as small. The physical pro- perties of the beam needed for the simple theory of flexure are then represented simply by the value of a certain quantity B which is called the flexural rigidity and which may be calculated when the form of the cross-section and the elasticity of the material of the beam are known. We are not interested at this stage in the calculation of B and shall conse- quently assume that the value of B for a given beam is known. The funda- mental hypothesis on which, the theory is based is that when the beam is bent by external forces there is at each point x of the central axis a resisting couple proportional to the curvature of the beam which just balances the bending moment introduced by the external forces. When the flexure takes place in the plane of xy this resisting couple has a moment which S+dS Bending of a Beam 17 can be set equal to JBd*y/dx2 and the fundamental equation for the bending moment is M = Bdty(da.tm The origin of the bending moment will be better understood when it is remarked that the bending moment M is associated with a transverse shearing force 8 by the equation - s = dM/dx. When the beam is so light that its Fig. 2. weight may be disregarded, this shearing force S is constant along any portion of the beam that does not contain a point of support or point of attachment of a weight. If we have a simple cantilever OA built into a wall at O and carrying a weight W at the point B the shearing force 8 is zero from A to B and is W from B to 0, while M is zero from A to B and equal to Wx between B and 0. At the point 0 the fact that the beam is built in or clamped implies that ?/== 0 and dy/dx = 0, consequently the equation Bd*y/dx* = - W x + W b gives y = - W x*/6B + Wbx2/2B. O B W -a- This holds for x < b. For x > b the differential equation for y is Bd*y/dx* = 0, and so y — mx + c. The quantities y and dy/dx are supposed to be continuous at B and so we have the equations mb + c = W b*/3B, m = PF62/25 which give c == — Wb3/6B. The deflection of B is Wb3/3B and is seen to be proportional to the force W, The deflection of A is also proportional to W. § 1-22. Let us next consider the deflection of a beam of length I which is clamped at both ends x = 0, x = I and which carries a concentrated load W at the point x = £. We have the equations S = I7 + ff, S = T, say, - M = (T+ W" )x + ^= -Bd2y/dx*, - M = Tx £ (T7 + IF) x2 + NX - - JWy/ete, ^T7^2 $(T+W)x*+ $Nx* = - By, W£ + N = - Bd2y/dx2, N)(x-l) = - Bdy/dx, (a: - /) 18 The Classical Equations where T and N are constants to be determined. M has been made con- tinuous at x = £, but we have still to make y and dy/dx continuous. This gives the equations \(W&- T/2)- Wlg + Nl, i (Wf» + TZ») = j (_ Pf + T) Pf + prj (2f _ i) + Therefore Tl* = W£* (2f - 3Z), ZW = Iff (2£Z - Z2 - (Z - £)2 [a; (Z + 2f) This solution will be written in the form By = — TFgr (#, f ) and the function gr (x, ^) will be calle.d a Green's function for the differential ex- pression d*yldx* and the prescribed boundary conditions. If By= it is found on differentiation that y is a solution of the differential equation the function w (x) being supposed to be continuous in the range (0, /). Thi^ solution corresponds to the case of a distributed load of amount wdx for a length dx. -When w is independent of x the expression found for y is in By=^x*(l-x)*. It should be noticed that the Green's function g (x, f ) is a symmetrical function of x and y, its first two derivatives are continuous at x = f , but the third derivative is discontinuous, in fact The reactions at the ends of the clamped beam with concentrated load are found by calculating the shear S. When x < £ we have S = W (I - ^)2 (Z + 2f )/Z», and this is equal in magnitude to the reaction at x == 0. The reaction at x = I is similarly R = W (31 - 2f) f 2/Z'. The deflection of the point x = f is % = W? (I - WIWB, when £ - Z/2 this amounts.to W7S/192J3. In the case of the uniformly loaded beam the reactions at the ends are respectively £ W and | W as we should expect. The deflection of the middle point is W Z4/384£. The Green's Functions 19 § 1*23. When the beam is pin -jointed at both ends, M is zero there and the boundary conditions are y = 0, d2y/dx2 = 0 for x = 0 and x = a. When there is a concentrated load IF at x = £ and the beam is of negligible weight, the solution is By — Wk (x, £), where & (x, f) = x (a - £) (x2 + £2 - = g(a- x) (x2 -f £2 - The reactions at the supports are J?0 = W (1 - |/a) at a = 0, J^a = W^/fl at a: = a. As before, the deflection corresponding to a distributed load of density w (x) is and when w is constant By = w (x* - 2ax* -f a3#)/24. The reactions at the supports are in this case 7? = wcl — n ./TO " o a u x — • v/j 2i T> W(l jKa — — at* x === a* In the case of a beam of length Z clamped at the end x = 0 and pin- jointed at the end x = Z, the solution for the case of a concentrated load W at x = | is The deflection at # = f is now y = Tf£3 (j _ ^2 (4jj _ |)/12£ when | = Z/2, and the reaction at # = Z is If, on the other hand, we consider a beam which is clamped at the end x = 0 but is free at the end x = Z except for a concentrated load P which acts there, we have, at the point x — £, while at the point x — I Byl = Hence R = Wyf:jyl . If the original beam is acted on by a number of loads of type W we have, for the reaction at the end x = Z, 20 The Classical Equations On account of this relation the curve (I) is called-the " influence line" of the original beam. Much use is made now of influence lines in the theory of structures*. There are three reciprocal theorems analogous to g (x, £) = g (£, x) which are fundamental in the theory of influence lines. These theorems, which are due to Maxwell and Lord Rayleigh, may be stated as follows: Consider any elastic structure with ends fixed or hinged, or with one end fixed and the other hinged, to an immovable support, then (1) The displacement at any point A due to a load P applied at any point B is equal to the displacement at B due to the same load P placed at A instead of B. (2) If the displacement at any point A is prevented by, a load P at A with displacement yB at B under a load Q, and alternatively if a load Ql at B prevents displacement at B with displacement yA at A under a load P, then if yA = yR , P must equal Ql . (3) If a force Q acts at any point B producing displacements yB at B and yA at any other point A, and if a second force P is caused to act at A but in the opposite direction to Q reducing the displacement at B to zero, then Q/P = yA/yB* In these three relationships it is supposed that the displacements are in the directions of the acting forces. Proofs of these relations and some applications will be found in a paper by C. E. Larard, Engineering, p. 287 (1923). § 1-24. Let us next consider a continuous beam with supports at A, B and C. The bending moment M at any point in AB or BC is the sum of bending moment Ml of a beam which is pin-jointed at ABC and of the moment M2 caused by the fixing moments at the supports. Let us take B as origin and let 12 denote the length BC. For a beam which is pin-jointed at B and C we have M1 = \w (I2x - x2), while for a weightless beam with fixing moments MB, Mc at B and C respectively, we have M2 = - MB - (Mc - MB) x/l2. Hence M = B2d2y/dx2 = \w2l2x - \w2x2 - MB - (Mc - MB) x/l2. Integrating, we have B2dy/dx = ±w2l2x2 - w2x*/$ - MBx - x2 (Mc - MB)/212 - B2iB, where IB is the value of dy/dx at x = 0. Integrating again, B2y = W2l2x3/l2 - w2x*/24: - x2MB/2 - x9 (Mc - MB)/6l2 - B2iBx. When x - 12, y = - y2, say 6B2iB = ±w2l2* - 2MB12 - MC12 + M^yz. * See especially Spofford, Theory of Structure*; D. B. Steinman, Engineering Record (1916); G. E. Beggs, International Engineering, May (1922). The Equation of Three Moments 21 Similarly, by considering the span BA, taking B again as origin, but in this case taking x as positive when measured to the left, Eliminating i& we obtain the equation B^ (MA + 2MB) + BJt (Mc + 2MB) - J (^A3S2 + wJfBJ + 6 (B2ll~lyl + B^l^y^. This is the celebrated equation of three moments which was given in a simpler form by Clapeyron* and subsequently extended for the general case by Heppelt, WeyrauchJ, Webb§ and others ||. The reaction at B is the sum of the shears on the two sides of B and is therefore 7 ,, ,, 7 ,, ,, % = W2*2 + MB- MC ^ wJi MR - MA Similarly for the other supports. § 1-25. When a light beam or thin rod originally in a vertical position is acted upon by compressive forces P at its ends (Fig. 4) | the equation for the bending moment is M = Bd2y/dx2 = - Py, or d2y/dx2 + kzy = 0, where k2 = P/B; and if y = 0 when x — 0 and when x = a, the solution is y = A sin kx, where sin ka = 0 or A = 0. If ak < 77, the analysis indicates that A = 0. A solution with A £ 0 becomes possible when ak = TT. The corresponding load P = B7T2/a2 is called Euler's critical load for a rod pinned at its ends. When P is given there is a corresponding critical lejijgth a = P^/B^Tr. To obtain these critical values experimentally great care must be taken to eliminate initial curvature of the rod and bad centering of the loads. The formula of Euler has been confirmed by the experiments of Robertson. In general practice, however, the crippling load PC is found to be less than the critical load P0 given by Euler's formula, and many formulae for struts have been proposed. For these reference must be made to books on Elasticity and the Strength of Materials. In the case of a strut clamped at both ends there is an unknown couple MQ acting at each end. The equation is now Bd*y/dx2+ Py=M0, * E. Clapeyron, Comptes Retidus, t. XLV, p. 1076 (1857). t J. M. Heppel, Proc. lust. Civil Engineers, vol. xix, p. 625 (1859-60). J Weyrauch, Theorie der continuierlichen Trdger, pp. 8-9. § R. R. Webb, Proc. Camb. Phil Soc. vol. vi (1886). (Case Bl 4= B2.) || M. Levy, Statique graphique, t. n (Paris, 1886). (Case yl 4= 0, yz 4s 0.) 22 The Classical Equations and the solution is of type Py = MQ -f a cos kx -\- j$ sin &#. The boundary conditions y = 0, dy/rfa = 0 at a: = a and # = 0 give 0=0, a = — MQ, MQ sin &a = 0, M0 (1 — cos ka) = 0. Hence either Jfef0 = 0 or sin (to/2) = 0. The critical load is now given by the equation ka = 2?r and is P0 = 4Brr2/a2. When the load P reaches the critical value the rod begins to buckle, and for a discussion of the equilibrium for a load greater than PQ a theory of curved rods is needed. In the case of a heavy horizontal beam of weight w per unit length and under the influence of longitudinal forces p at its ends, the equation satisfied by the bending moment M is where k* = P/B. If M = M0 when x = 0 and M = Mj when a; = a, the solution of the differential equation is v^ — M) sin ka = ( »2 "" ^o ) sin ^ (a ~" x) + ( 7T2 ~ -^i ) s*n ^ Let us assume that MQ and Jf 1 are both positive and write M 0 = , 0 sin2 0, Jkf i = , 0 sin2 ^, kx = a. k (a — x) — B, ka = a + B. fc2 k* r The equation which determines the points of zero bending moment (points of inflexion) is sin (a -f 0) = cos 20 . sin j8 -f cos 2(f> . sin a. We shall show that if a and 0 are both positive this equation implies that a-f/3> 2 (# + </>) and so determines a certain minimum length which must not be exceeded if there are to be two real points of inflexion. Let us regard 6 and <f> as variable quantities connected by the last equation and ask when 0 + </> is a maximum. Writing z = B -f <f> we have -T| = 1 + -~f , 0 = sin 28 sin B -f- sin 2<A sin a ^ , au au au *z d*<f> 2sinj8 f b"2 = "^« ^ ~ ~ - ^T7T7 c O* dO* sm a sm2 2(f>[_ ft rt/ . cos 2 sm 2 - sin When 2$ = a, we have 2<f> = j8, and these values of 6 and <£ give a zero value of JQ and a negative value of ^ , they therefore give us a maximum value of 2, and so for ordinary values of 6 and <f> we have the inequality 2 (0 + <£) < a -f j8. Stability of a Strut 23 The position of the points of inflexion is of some practical interest because, in the first place, A. R. Low* has pointed out that instability is determined by the usual Eulerian formula for a pin- jointed strut of a length equal to the distance between the points of inflexion, if these lie on the beam, and secondly, if any splicing is to be done, the flanges should 'be spliced at one of the points where the bending moment is zerof. When P is negative and so represents a pull we may put p2 = - P/B, and the solution is ( ~2 ~^~ M } sinh pa =» ( — -f MQ } sinh p (I — x) -f ( 0 4- M* ) sinh vx. \p* / \p£ ) N \p2 LJ r If we write 2w ,- 2w M0 = — sinh- 0, MI "- sinh2 ^6, px = a, p (a - x) = )8, pa = a + fi, p p a value of x for which M == 0 is determined by the equation sinh (a -f- jS) = cosh 20 sinh ^3 -f cosh 2^> sinh a. This equation implies that a -f j3 > 2 (0 -f (/>). For a continuous beam acted on by longitudinal forces at the points of support there is an equation analogous to the equation of three moments which is obtained by a method similar to that used in obtaining the ordinary equation of three moments. We give only an outline of the analysis. Case 1. k*=P2/B2, EG = 6, 2/3 = bk, yA = VB = yc =* 0, where * .4ero?iaMftcaZ Journal, vol. xvin, p. 144 (April, 1914). See also J. Perry, Phil Mag. (March, 1892); A. Morley, ibid. (June, 1908); L. N. G. Filon, Aeronautics, p. 282 (Sept. 1919). t H. Booth, Aeronautical Journal, vol. xxiv, p. 563 (1920). 24 The Classical Equations There is a corresponding equation for the bay BA which is of length a, if h2 = Pi/Bi , 2a = ah, the equation of three moments has the form aMA bMc f,n\ , OTIT ! a JL t \ _. * JL /m' ^ia3 / / \ _L -^-/(a)-f -g- /(/?) -f 2^ j^ '0 («)+ ^"^(P)f = 4^ ^(a) + Case 2. P < 0. The corresponding equations are Pj/JJj, 2a = ah, 2/? = 6i, O (a) + I O (j8)J r, . . 3 1 - 2a cosech 2a A , . 3 2a coth 2a — 1 *•<«) = 2-~ --, - -, OW-i - ^i— • V (a) = The functions/ (a), P (a), etc., have been tabulated by Berry* who has also given a complete exposition of the analysis. These equations are much used in the design of airplanes built of wood. EXAMPLES 1. Find the crippling load for a rod which is clampod at one end and pinned at the other. 2. Prove that jn the case of a uniform light beam of length a with a concentrated load W at x = | the solution can be written in the form « 2Wa I . nx . TT£ 1 . 2nx . 2n{ \ M = - 1 sin — sin --- h ^ sin — sin +...), 7T2 \ a a 22 a a ) TTX . TT£ 1 a~ 8m a + 2* when the beam is pinned at both ends. The corresponding formulae for a uniformly dis- tributed load are , . 4tW f . TTX 1 . 3irX 1 . 5nX \ w(x) = - (sin— -f Osm ---- 1- _ sin +...), TT \ a 3 a 5 a ) ,. M = va2 ( . nx 1 . 3nx 1 . 57ra; \ -„- sm ---- h jr- sin --- h ^> sin --- h ... ) , T3 \ a 33 a 53 a / 4wa* f . TTX 1 . STTX 1 . STTX \ y = D . sin --- h 5- sm — -f- K5 sin --- -f ... . Bn5 \ a 3° a 55 a / [Timoshenko and Lessells Applied Elasticity, p. 230.] 3. Find the form of a strut pinned at its ends and eccentrically loaded at its ends with compressional loads P. 4. The Green's function for the differential expressionf * Trans. Roy. Aeronautical Soc. (1919). The tables are given also inPippard and Pritchard*a Aeroplane Structures, App. I (1919). f Examples 4-6 are taken from a paper by A. Myller, 2$as. Oottingen (1906). End' Conditions 25 and the end conditions u (0) - u (I) = u' (0) - u' (1) - 0 is ~ fl\*-\ &>• *M- ^ «_ [l*jk (z)> P- Jo £<*")' where 5. If in the last example the boundary conditions are u (0) = u' (0) = u" ( 1 ) = u"' ( 1 ) = 0 the Green's function is * ' * ~ 4 6. When the end conditions are u (0) = w" (0) = u (1) =. u" (1) = 0, the Green's function is 0 (#, ( ), where - (a - 4)3 + 4y)*f - (0 - 2y)(x + f ) - y. § 1-31. jFVee undamped vibrations. Whenever a particle performs free oscillations in a straight line under the influence of a restoring force pro- portional to the distance from a fixed point on the line the equation of motion is mx = - where m is the mass of the particle and X\L is the restoring force. Writing M = 4«m, we have £ an equation which has already been briefly considered. The general solution is gn where ^4 and JS are arbitrary constants. Writing k = 27m, A = a sin B = a cos 0 we have x = a sin (27m£ -f 0). The quantity a specifies the amplitude, n the frequency and 27fnt -f 0 the phase of the oscillation. The angle 0 gives the phase at time t = 0. The period of vibration T may be found from the equations kT - 277, nT = 1. This type of vibration is called simple-harmonic vibration because it is of fundamental importance in the theory of sound. The vibrations of solid bodies which are almost perfectly rigid are often of this type, thus the end of a prong of a tuning fork which has been properly excited moves in a manner which may be described approximately by an equation of this 26 The Classical Equations type. The harmonic vibrations of the tuning fork produce corresponding vibrations in the surrounding air which are of audible frequency if 24 < n < 24000. The range of frequencies used in music is generally 40 < n < 4000. The differential equation (I) may be replaced by two simultaneous equations of the first order x + ky=0, y-kx=0 (II) which imply that the point Q with rectangular co-ordinates (x, y) moves in a circle with uniform speed ka. We have, in fact, the equation xx 4- yy = 0, which signifies that x2 -f- y2 is a constant which may be denoted by a2. There is also an equation #2 -f I/* = k* (3.2 _j_ y2) = £2a2? which indicates that the velocity has the constant magnitude ka. The solution of the simultaneous equations may be expressed in the form x = a cos a, y = a sin a, where a — Zrrnt 4- 6 — ~ ; • Simultaneous equations of type (II) describe the motion of a particle which is under the influence of a deflecting force perpendicular to the direction of motion and proportional to the velocity of the particle. The equations of motion are really x -f- ky — 0, y — kx = 0, but an integration with respect to t and a suitable choice of the origin of co-ordinates reduces them to the form (II). The equations may also be written in the form 7 , u + kv = 0, v — leu = 0, where (u, v) are the component velocities. If the deflecting force mentioned above is the deflecting force of the earth's rotation the deflection is to the right of a horizontal path in the northern hemisphere and to the left in the southern hemisphere. If the angle <f> represents the latitude of the place and o> the angular velocity of the earth's rotation, the quantity k is given by the formula k — 2a> sin (f>. When the resistance of the air can be neglected, the suspended mass M of a pendulum performs simple harmonic oscillations after it has been slightly displaced from its position of equilibrium. The vertical motion is now so small that it may be neglected and the acceleration may, to a first Simple Periodic Motion 27 approximation, be regarded as horizontal and proportional to the horizontal component of the pull P of the string. We thus have the equation of motion Mix = - Px = - Mgx, where / is the length of the string and g the acceleration of gravity. The mass of the string is here neglected. With this Q simplifying assumption the j endulum is called a simple pendu- lum. In dealing with connected systems of simple pendulums it is convenient to use the notation (I, M) for a simple pendu- lum whose string is of length I and whose bob is of mass M (Fig. 5). If the string and suspended mass are replaced by a rigid body free to swing about a horizontal axis through the point 0, the equation of motion is approximately 19 = - MgliO, where 7 is the moment of inertia of the body about the horizontal axis through 0 and h is the depth of the centre of mass below the axis in the equilibrium position in which the centre of mass is in the vertical plane through 0. Writing Mhg = Ik2 the equation of motion becomes 0 + ]*0 - 0, and the period of vibration is 27r/k, a quantity which is independent of the angle through which the pendulum oscillates. This law was confirmed experimentally by Galileo, who showed that the times of vibration of different pendulums were proportional to the square roots of their lengths. The isochronism of the pendulum for small oscillations was also discovered by him but had been observed previously by others. When the pendulum swings through an angle which is not exceedingly small it is better to use the more accurate equation 0 + £2 sin 0=0, which may be derived by resolving along the tangent to the path of the centre of gravity G or by differentiating the energy equation = Mgh (cos 0 - cos a), which is written down on the supposition that the velocity of 0 is zero when 0 = a. With the aid of the substitution sin (|0) = sin (Ja) sin</>, this equation may be written in the form ^2^ p [i _ sin2 \a sin2 <£]. As 0 varies from — a to a, <f> varies from — « ^° 5 > an(^ so 28 The Classical Equations swing from one extreme position (0 = — a) to the next extreme position (0 = «) is 2 (1 - sin2 \a sin2 </»)"* d*j>. When a is small the period T is given approximately by the formula kT = 27r(l -f Jsin2 J«) and depends on a, so that there is not perfect isochronism. This fact was recognised by Huygens who discovered that perfect isochronism could theoretically be secured by guiding the string (or other flexible suspension) with the aid of a pair of cycloidal cheeks so as to make the centre of gravity describe a cycloidal instead of a circular arc. This device has not, however, proved successful in practice as it introduces errors larger than those which it is supposed to remove*. More practicable methods of securing isochronism with a pendulum have been described by Phillipsf. § 1-32. Simultaneous equations of type Lx -j- My -f Lm2x = 0, MX + Nij -f Nn*y = 0, in which L, M, Ny my n are constants, occur in many mechanical and electrical problems. When the coefficient M is zero the co-ordinates x and y oscillate in value independently with periods Sir/m and 27r/n respectively, but when M =£ 0 the assumption x = p2MAelpt, y = LA (m2 - p2) eipt gives the equation p* (1 - y2) - p2 (m2 4- n2) + m2n2 = 0, where y2 = M2/LN. This quantity y is called the coefficient of coupling! . When m =5^ ?i we have - „ . V2D4 7)2_m2== Yf * p2-n2' and when y is small the value of p which is close to m is given approximately by the equation >p2 _ m2 = -JT™ _ p 2 say. * m2- n2 r > j A simple harmonic oscillation of the #-co-ordinate, with a period close to the free period 2-jT/m, is accompanied by a similar oscillation of the * See R. A. Sampson^ article on " Clocks and Time-Keeping" inDictionary of Applied Physics, vol. m. t Comptes Rendus, t. oxn, p. 177 (1891). J See, for instance, E. H. Barton and H. Mary Browning, Phil. Mag. (6), vol. xxxiv, p. 246 (1917). Simultaneous Linear Equations 29 y-co-ordinate with the same period but opposite phase. The amplitude of the ^-oscillation is proportional to y2. Now let p1 be the greater of the two values of p. If m > n we have pl > m but if m < n we have p2 < m. The effect of the coupling is thus to lower the frequency of the gravest mode of vibration and to raise the frequency of the other mode of simple harmonic vibration. If m = n the equation for p2 gives p2 = m2 ± yp2, and the effect of the coupling is to make the periods of the two modes unequal. In the general case we can say that the effect of the coupling is to increase the difference between the periods. The periods may, in fact, be represented geometrically by the following construction : Let OA, OB represent the squares of the free periods, the points 0,A,B being on a straight line. Now draw a circle F on AB as diameter and let a larger concentric circle cut the line OA B in U and V \ the distances OC7, O V then represent the squares of the periods when there is coupling. If a tangent from 0 to the circle F touches this circle at T and meets the larger circle in the points M and L the coefficient of coupling is represented by the ratio TL/TO (Fig. 6). So long as 0 lies outside the larger circle it is evident that the difference between the periods is increased by the coupling, but when y > 1 the point 0 lies within the larger circle and the difference between the periods de- creases to zero as the radius CU of this circle increases without limit. There is thus some particular value of the coupling for which the difference between the periods has the original value, both periods being greater than before. When y = 1 the equations of motion may be written in the forms Lx + My + Lm2x = 0, Lx + My + Mn2y = 0, and imply that Lm2x = Mn2y. There is now only one period of vibration. The cases y> \ are not of much physical interest as the values of the constants are generally such that M 2 < LN, this being the condition that the kinetic energy may be always positive. Equations of the present type occur in electric circuit theory when resist- ances are neglected. In the case of a simple circuit of self-induction L and lg* capacity C furnished, say, by a Ley den jar in the circuit, the charge Q on the inside of the jar fluctuates in accordance with the equation ® = 0 30 The Classical Equations when the discharge is taking place. The period of the oscillations is thus T = 2n W(LC). This is the result obtained by Lord Kelvin in 1857 and confirmed by the experiments of Fedderson in 1857. The oscillatory character of the discharge had been suspected by Joseph Henry from observations on the magnetization of needles placed inside a coil in a discharging circuit. In the case of two coupled circuits (Ll9 C^), (L2, (72) the mutual induction M needs to be taken into consideration and the equations for free oscil- lations are ~ L& + MQ, + / - 0, § 1-33. The Lagrangian equations of motion. Consider a mechanical system consisting of / material points of which a representative one has mass m and co-ordinates x, y, z at time /. Using square brackets to denote a summation over these material points, we may express d'Alembert's principle in the Lagrangian form [m (xSx + y8y -f 5 82)] - [XSx + Y8y + ZSz], where 8x9 §//, 8z are arbitrary increments of the co-ordinates which are compatible with the geometrical conditions limiting the freedom of motion of the system. On account of these conditions, the number of degrees of freedom is a number N9 which is less than 3J, and it is advantageous to introduce a set of "generalised11 co-ordinates ql9 g2, ... gv which are inde- pendent in the sense that any infinitesimal variation 8qs of qs is compatible with the geometrical conditions. These conditions may, indeed, be expressed in the form #•=/(?! >?2» ••• <LV> 0> y = 9 (?1>?2> ••• 1\'>0> * = M<?1,?2> •'• <7A>0- Using the sign S to denote a summation from 1 to N, a prime, to denote a partial differentiation with respect to t and a suffix s to denote a partial differentiation of x9 y or z with respect to qs, we have the equations x = x' -f Zxsq89 Sx = I<x38qS9 where the quantities Q(s) may be called generalised force components associated with the co-ordinates q. The first of these equations shows that xs is also the partial derivative of x with respect to qs and so if the kinetic energy of the system is T9 where 2T = [m (x2 4- y2 + z2)], we shall have -_ Cl r Lagrange's Equations * 31 where p8 is the generalised component of momentum. Since dx Sxf v . dxr Sq^Wr^ "Xr*qs^~dt =s*r' and -.-- (xxs) = xxs -f #xg, rt* we have [m (xSx + ySy + 282)] - S8gs , m (^ + yys + 22,) - m (xxs -f yy, + zzs) iai . J and so Lagrange's principle may be written in the form dT\ v-/ir<,^ -** *9" On account of the arbitrariness of the increments 8qs this relation gives the Lagrangian equations of motion dfST^dT^ dt\dqs) dqs ^ ' If there is a potential energy function F, which can be expressed simply* in terms of the generalised co-ordinates </, we may write and the equations of motion take the simple form d i dL\ dL The quantity L is called the Lagrangian function. Introducing the reciprocal function T-x(adT] T 7-M?*a$J ' , dT _ f . d /ST\ .. dT) „ /.. ST . dT\ we have -- = S , L < ,-.- - S U + - Hence we have the energy equation T7 -f F = constant. When the functions /, g and h do not contain the time explicitly we have on account of Euler's theorem for homogeneous functions The Lagrangian equations of motion may be replaced by another set of equations for the quantities p and q. For this purpose we introduce the Hamiltonian function H defined by = - L+ ^p8qB- 32 The Classical Equations If we always consider H as a function of the quantities q8 and p8 but L as a function of q8 and qa, we have Thus *H-=q., m=-^~, consequently the equations of motion can be expressed in the Hamiltonian ^_ dt ~~ dp, ' dt ~ dq, * Systems of equations whose solutions represent superposed simple harmonic vibrations are derived from the Lagrangian equations of motion of a dynamical system d /dT\ _ dT __ _ SV dt \dqj dqs ~ dqs ' 5= 1,2, ... AT, whenever the kinetic energy T, and the potential energy V, can be ex- pressed for small displacements and velocities in the forms N N 2T= S 2 amnqmqn, m-l n-1 N N 2V = S S cmn?m?n 7/1-1 71—1 respectively, where the constant coefficients amn and cmn are such that T and F are positive whenever the quantities qm, qm do not all vanish. For such a system the equations (A) give the differential equations N £ Knrtfn + C«in?») ^ °> n-1 m- 1, 2, ... N. Multiplying by um, where um is a constant to be determined, and summing with respect to m, the resulting equation is of type v + k*v= 0, (B) if the quantities um are such that for each value of n N N S umcmn - fc2 S wroaww - P6n, say. ?n«l m-l The corresponding value of v is then N v= S 6n?n. n-l Normal Vibrations 33 Now when the quantities um are eliminated from the linear homo- geneous equations N 2 (cmn - k2amn) um = 0, (C) ?n = l we obtain an algebraic equation of the A7th degree for k2. With the usual method of elimination this equation is expressed by the vanishing of a determinant and may be written in the abbreviated form I c — k2n I — 0 I ^mn K amn \ — u» To show that the values of k2 given by this equation are all real and positive, we substitute k2 = h -f ij, um = vm -f iwm in equation (C). Equating the real and imaginary parts, we have N N 2 (cmn - Juimn) vm + j 2 amnwm = 0, m — 1 m — 1 AT iv 2 (cmn - Aamn) wn-j 2 amnt?w - 0. ra = 1 ?7i — 1 Multiplying these equations by wn and — vn respectively, adding and summing, we find that j 2 amn (wmwn -f vmvn) = 0. in, n The factor multiplying j is a positive quadratic form which vanishes only when the quantities vn, wn are all zero, hence we must have j — 0 and this means that k2 is necessarily real. That k is necessarily positive is seen immediately from the equation 2 cmnumun = k2 2 amnumun, m, n m,n which involves two positive quadratic forms. If um (A^), um (k2) are values of um corresponding to two different values of k we have the equations 2 (Cmn - *!«««») Um (*i) = 0, 7/J-l N S (cwn - k22amn) um (k,) = 0. TO-l Multiplying these by wn (A:2), wn (^2) respectively and subtracting we find that (V _ V) s a^Wm (Aj) ^ (kj = Q (D) m,n Denoting the constant 6W associated with the value k by 6n (^), we see from the last equation that if k2 =£ kl9 S 6n (*i) ^n (*i) = 0. n-1 On the other hand, N 2 6n fa) un (k^ = 2 amnum (*J wn (^), n» 1 m, n 34 The Classical Equations and is an essentially positive quantity which may be taken without loss of generality to be unity since the quantities un (k^ contain undetermined constant factors as far as the foregoing analysis is concerned. Using the symbol v (k, t) to denote the function v corresponding to a definite value of k, we observe that if N qm= S v(k8,t) Ams, 8-1 N we must have S bn (k3) \ms = 0 n±m s-l = 1 n = ra. Multiplying by un (kr) and summing with respect to n we find that *mr = Um (kr). N Hence qm= I, um (ks) v (ks, t). .9-1 This expresses the solution of our system of differential equations in terms of the simple harmonic vibrations determined by the equations of type (B). The analysis has been given for the simple case in which the roots of the equation for k2 are all different but extensions of the analysis have been given for the case of multiple roots. The relation (D) may be regarded as an orthogonal relation in general- ised co-ordinates. When amn= 0, m±n, amm= 1, the relation takes the simpler form .v 2 um (kv) um (kq) = 0 p + q. m -1 § 1-34. An interesting mechanical device for combining automatically any number of simple harmonic vibrations has been studied by A. Gar- basso*. A small table of mass m is supported by four light strings of equal length / so that it remains horizontal as it swings like a pendulum. The table is attached at various points to n simple pendulums (ls, ras), s = 1, 2, ... n. Each string is regarded as light and is supposed to oscillate in a vertical plane and remain straight as the apparatus oscillates. Specifying the configuration of the apparatus by the angular variables $o> ^i > ••• ®n we have in a small oscillation V = * Vorlesungen uber Speklroskopie, p. 65; Torino Atti, vol. XLIV, p. 223 (1908-9). The case in which n = 2 has been studied in connection with acoustics by Barton and Browning, Phil. Mag. (6), vol. xxxiv, p. 246 (1917); vol. xxxv, p. 62 (1918); vol. xxxvi, p. 36 (1918) and by C. H. Lees, ibid. vol. XLVHI (1924). Compound Pendulum The equations of motion are ms 8=1 n \ 2 ms } 00 = 0, -1 / 35 (8= 1,2, ...n). n Writing ma = csm0, S cs = c, the equation for k2 is in this case + C) - -/0P - I0kz - gc, - gc2 ... - gcn | =o, -Z^ o ... 0 | 0 g-l2k*... 0 I 0 or g- 0 Expanding the determinant we obtain the equation where / (V) = II (g - l,k*). .s-1 Now (10 - ls) [g - (10 + ls) k*} = la(g- I0lc*) -l,(g- l,k*), consequently the equation for k2 can be written in the form /(*») \(g - kk*) fl + gk 2 j- - r~f— -rJ - g S C8's 1 = 0. ( L S-l "O ~ 18) \(J — LS^ )J 8-1 ^0 ~ ^S) If the mass of each pendulum is so small in comparison with that of the table that we may neglect terms of the second order in the quantities cs, the equation may be written in the form (g - U« - 17 S ±l> } 0 (g - l,k* + ^-) . \ .s-1 ^0 ~ 1J .s-1 \ ^0 ~~ 's/ Hence the periods of the normal vibrations are approximately - 1,2, ...n). 3-2 36 The Classical Equations If IQ > 18 the period of the 5th pendulum is decreased by attaching it to the table. If 10 < ls the period is increased. EXAMPLES 1. A simple pendulum (6, N) is suspended from the bob of another simple pendulum (a, M ) whose string is attached to a fixed point. Prove that the equations of motion for small oscillations are ^Tv „.. ^T •• (M + N) a2d + Nab<f> + (M + N) gaS = 0, Nab'B + Nb*j> + Ngb<f> = 0, where B and <£ are the angles which the strings make with the vertical. 2. Prove that the coefficient of coupling of the compound pendulum in the last example is given by _ ___ M + N' 3. Prove that it is not possible for the centre of gravity of the two bobs to remain fixed in a simple type of oscillation. 4. A simple pendulum (/, 'M) is suspended from the bob of a lath pendulum which is treated as a rigid body with a moment of inertia different from that of the bob. Find the equations of motion and the coefficient of coupling. 5. A simple pendulum (I, M) is attached to a point P of an elastic lath pendulum which is clamped at its lower end and carries a bob of mass N at its upper end. At time t the horizontal displacements of M, N and P are y, z and az respectively, a being regarded as constant. By adopting the simplifying assumption that a horizontal component force F at P gives N the same horizontal acceleration as a force aF acting directly on N, obtain the equations of motion IMy + Mg (y - az) - 0, INz + lNn*z = Mga (y - az), and show that the coefficient of coupling is given by the equation a? ~ Nri* + Mm**9 where lm2 = g. [L. C. Jackson/PAtf. Mag. (6), vol. xxxix, p. 294 (1920).] ft. Two masses m and m' are attached to friction wheels which roil on two parallel horizontal steel bars. A third mass J/, which is also attached to friction wheels which roll on a bar midway between the other two, is constrained to lie midway between the other two masses by a light rigid bar which passes through holes in swivels fixed on the upper part of the masses. The masses m and m' are attached to springs which introduce restoring forces proportional to the displacements from certain equilibrium positions. Find the equations of motion and the coefficient of coupling. This mechanical device has been used to illustrate mechanically the properties of coupled electric circuits. [See Sir J. J. Thomson, Electricity and Magnetism, 3rd ed. p. 392 (1904); W. S. Franklin, Electrician, p. 556 (1916).] 7. Two simple pendulums (/lf Jfj), (/2, M2) hang from a carriage of mass M which, with the aid of wheels, can move freely along a horizontal bar. Prove that the equations of motion are (M + M1 + M2)x + MJ& + M212§2 = 0, Quadratic Forms 37 Hence show that the quantities B19 92 can be regarded as analogous to electric potential differences at condensers of capacities M^g and M2g, the quantities Mlllgdl and Mzl2g6z as analogous to electric currents in circuits, the quantities , ) as analogous to coefficients of self-induction and [(Ml + M2 + M ) g2]~* as analogous to a coefficient of mutual induction. [T. R. Lyle, Phil. Mag. (6), vol. xxv, p. 567 (1913).] 8. A simple pendulum of length Z, when hanging vertically, bisects the horizontal line joining the knife edges. When the pendulum oscillates it swings freely until the string comes into contact with one of the knife edges and then the bob swings as if it were suspended by a string of length h. Assuming that the motion is small and that in a typical quarter swing 10 + go = 0 for 0 < t< T, hd + 00 = 0 for r < t < T, prove that the quarter period T is given by the equation m cot n (T — r) = n tan mr, where g = Im? = hri2. § 1-35. Some properties of non-negative quadratic forms*. Let nt n g = 2 grsxrxs i, i be a quadratic form of the real variables xl9 ... xn, which is negative for no set of values of these variables, then there are n linear forms 1 with real coefficients prg such that n / n g = S ^ 2 prsx, This identity gives the relation n glk = S ^rt^rfc i which can be regarded as a parametric representation of the coefficients in a non-negative form. This result may be obtained by first noting that gss is not negative, for g^ is the value of g when xr = 0, r £ s and xs = 1 . If the coefficients gr9 are not all zero the coefficients gss are not all zero, because if they were and if, say, 012 <0 a negative value of g could be obtained by choosing x1 = 1, x2 = T 1, #3 = ... o:n = 0. We may, then, without loss of generality assume that there is at least one coefficient gu of the set g88 which is positive. * L. Fejer, Math. Zeits. Bd. i, S. 70 (1918). 38 The Classical Equations Writing pnp18 = gls s = 1, 2, ... n, n 2j = S plsXs, 8-1 <7<" = g - «,», it is easily seen that the quadratic form g(l} does not depend on xv g(l} is moreover non-negative because if it were negative for any set of values of #2, #3, ... xn we could obtain a negative value of g by choosing xl so that 2l = 0. ' Since g(l) is non-negative it either vanishes identically or the coefficient of at least one of the quantities #22, x32, ... xn2 in f/{1) must be positive. Let us suppose that gr22(1) is positive and write (1> r = 2, 3,...tt/ 0(2) _ yd) _ z22. Continuing this process it is found that g = 2^+ z224- . . .zn2, where the linear forms ^ , z2, . . . zn are not all zero ; it is also found that none of the quantities gn, <722(1), fe*2*, ... are negative and that all these quantities, except the first, are ratios of leading diagonal minors in the determinant | 9rs | , and are not all zero. n, n Now let h = £ htkxtxk i, i be a second non-negative form, and let n hik = S Jat^fc be its parametric representation, then* n, n n,n n 2 gtkhtk= 2 ( S 1,1 1,1 \<r-l If 0> 2/i, ^2» ••• 2/n are arbitrary real quantities, 2 (x, - 6ysy is never negative. Regarding this as a quadratic expression in 0 it is readily seen that the quadratic form A~S*,*£ys»-(£*.y.)» i i i is non-negative. This result, which was known to Cauchy and Bessel, is frequently called Schwarz's inequality as Schwarz obtained a similar inequality for integrals. * L. Fej£r, Math. Zeits. Bd. i, S. 70 (1918). Hermitian Forms 39 Using this particular form of h in Fejer's inequality we obtain the result that n, n n n Xgrsyry,<X9rrXy*. 1,1 11 For further properties of quadratic forms the reader is referred to Brom- wich's tract, Quadratic Forms, Cambridge (1906), to Bocher's Algebra, Macmillan and Co. (1907), and to Dickson's Modern Algebraic Theories, Sanborn and Co., Chicago (1926). § 1-36. Hermitian forms. Let z denote the complex quantity conjugate to a complex quantity z and let the complex coefficients crs be such that n,n the bilinear form 2 crszrzs i, i is then Hermitian, If clm = alm + iblm,zm = rmelQm, where alm,blm, rm, 9m are real quantities, we have alm ~ aml> bim = — Omi, and the Hermitian form can be expressed as a quadratic form n,n 2 ptmrirm, i, i where plm = alm cos (0t - 6J + blm sin (0t - 0m) = pml. The positive definite Hermitian forms which are positive whenever at least one of the quantities zl , z2 , . . . zn is different from zero are of special interest. In this case the associated quadratic form is positive for all non- vanishing sets of values of rl , r2 , . . . rn and for all values of 01 , 62l ... 6n. An important property of a Hermitian form is that the associated secular equation | crs - A8r. | = 0 8r3 = 1 r = s - 0 r + s has only real roots. When the form is positive these roots are all positive. The proof of this theorem may be based upon analysis very similar to that given in § 1-33. EXAMPLES 1. If F(t) ^ 0 for - TT ^ * ^ TT and = 2 c¥eM, \> =s ~ 00 n, n #n= 2 Cl-mZlZm n» 1,2,3..., then #n ^ 0. This has been shown by Carath£odory and Toeplitz to be a necessary and sufficient condition that F(t) > 0. See Rend. Palermo, t. xxxn, pp. 191, 193 (1911). 40 The Classical Equations 2. If /(*)= f° eiteo> (x)dx, J — oo where o> (a;) = to (— #) n, n _ ftnd #„= 2 a) (a?j - *m) fzfm, Mathias has shown that when f(t) ^ 0 we have Hn ^ 0 for any choice of real parameters xl , x2 , . . . xn and of the complex numbers f, , f, , . . . fn . See Jlf ath. Zeite. Bd. xvi, p. 103 ( 1923). The analysis depends upon Fourier's inversion formula which is studied in § 3-12 and it appears that, with suitable restrictions on the function <*>(x), the inequality Hn ^> 0 is the necessary and sufficient condition that f(t) ^ 0. Mathias gives two methods of choosing a function a>(x) which will make Hn ^ 0. The correctness of these should be verified by the reader. (1) If the functions x(x)> x(x) are such that when x has any real value x(x) an(l X (x) are conjugate complex quantities, the function /«> X(a+ %)\(a - x)da -00 will make Hn ^ 0. (2) If the positive constants \v and the functions xv(x) are such that is a function of x — y> say o> (x — y)y then this function o> (x) will make Hn ^ 0. § 1-41. Forced oscillations. When a particle, which is normally free to oscillate with simple harmonic motion about a position of equilibrium, is acted upon by a periodic force varying with the time like sin (pt), the equation of motion takes the form x 4- k2x = A sin pt. Writing x = z -f- C sin pt, where G is a constant to be determined, we find that if we choose C so that (k*-p*)C = A, ...... (A) the equation for z takes the form z -f k*z = 0. The motion thus consists of a free oscillation superposed on an oscilla- tion with the same period as the force. In other words the motion is partly original and partly imitation. It should be noticed, however, that if p* > k2 the imitation is not perfect because there is a difference in phase. The difference between the case p* > k2 and the case p2 < k2 is beautifully illustrated by giving a simple harmonic motion to the point of suspension of a pendulum. When p2 == k2 the quantity C is no longer determined by equation (A) and the solution is best obtained by the method of integrating factors which may be applied to the general equation x + k2x = F (t). Forced Oscillations 41 Multiplying successively by the integrating factors cos kt and sin kt and integrating, we find that if x = c, x = u when t = T, we have ft x cos kt -f fc sin kt = u cos &T -f &c sin kr + \ F (s) cos ks . d#, r< # sin kt — kx cos kt — u sin &r — kc cos &r + \ F (s) sin &s . ds, rt kx = u sin A; (t — r) -f &c cos k (t — r) -f F (s) sin &($ — *) ds, ct x = u cos A: (t — r) — &c sin k (t — r) -f I F (s) cos k (t — s) ds, J T the function F (s) being supposed to be integrable over the range T to t. In particular, if the particle starts from rest at the time t we have at any later time t kx = IF (s) sin k (t — s) ds, rt x = IF (s) cos k (t — s) ds. J r When F (s) = A sin ks and r = 0 we find that 2fcr = t cos itf — &"1 sin kt, and the oscillations in the value of x increase in magnitude as t increases. This is a simple case of resonance, a phenomenon which is of considerable importance in acoustics. In engineering one important result of resonance is the whirling of a shaft which occurs when the rate of rotation has a critical value corresponding to one of the natural frequencies of lateral vibration of the shaft. For a useful discussion of vibration problems in engineering the reader is referred to a recent book on the subject by S. Timoshenko, D. Van Nostrand Co., New York (1928). By choosing the unit of time so that k = 1 the mathematical theory may be illustrated geometrically with the aid of the curve whose radius of curvature, p, is given by the equation p = a sin OHJJ, where ^ is the angle which the tangent makes with a fixed line. Using p now to denote the length of the perpendicular from the origin to the tangent, we have the equation ^ ...... (B) The quantity p thus represents a solution of the differential equation, and by suitably choosing the position of the origin the arbitrary constants in the solution can be given any assigned real values. In this connection d*D it should be noticed that ~j has a simple geometrical meaning (Fig. 7). When co = 1 the equation (B) is that of a cycloid, while epicycloids and hypocycloids are obtained by making a) different from unity. The intrinsic equation of these curves is in fact 4 (a -f b) b . aJj Q — V ' „ am ..... T ___ o — - OIJ.JL rt, , a a + 2b 42 The Classical Equations where a is the radius of the fixed circle and b the radius of the rolling circle which contains the generating point. Epicycloids 6 > 0. Pericycloids and hypocycloids 6 < 0. Fig. 8. § 1-42. The effect of a transient force in producing forced oscillations is best studied by putting r = — oo and assuming that c and u are both zero, then Too kx = F (t — cr) sin ka . da, Jo f00 x = F (t — a) cos ka . da. ' Let us consider first of all the case when In this case F' (t) is discontinuous at time t = 0 in a way such that F' (- 0) - F' (-f 0) - 2h. The solution which is obtained by supposing that x, x and x are con tinuous at time t = 0 is ft Too lex = e-w-*> sin kada -f .'0 J« 2 A sin let k*+h* _ht . _ 2h cos kt — he~ht £ == ~v"o ; »~ o ~ i > 0 It will be noticed that x is discontinuous at t = 0. Residual Oscillation 43 It is clear from these equations that x increases with t until t reaches the first positive root of the transcendental equation 2 cos kt = e~ht. The corresponding value of x is then 2 (h sin kt -f k cos kt) As t increases beyond the critical value x begins to oscillate in value, and as t -> oo there is an undamped residual oscillation given by _ 2h sin kt A general formula for the displacement x at time t in the residual oscillation produced by a transient force may be obtained by putting t = oo in the upper limit of the integral (t being retained in the integrand) *. This gives kx = I F (s) sin k (t — s) ds. J -00 In particular, if ,, , x [b 7 sin bs F (s) = cos ms . dm = — - , Jo s we have /*QO f? Q r °^ ft ^ kx — sin kt cos ks sin bs cos kt sin ks sin 65 ; .' -oo S J _oo S the second integral vanishes arid we may write TOO • ^£ 2&# = sin kt [sin (ft -f k) s -f sin (6 — k) s] — J -oo S = 277 sin &£ if 6 > & > 0 ' = 0 if 0 < b < k = TT sin kt if b = & > 0. There is thus a residual oscillation only when b > k. EXAMPLES /•& 1. If F (s) = I , cos m* . dm, there is a residual oscillation only when k lies within the range a < &< 6. Extend this result by considering cases when F (8) = / cos ms . <f> (m) dm, F (s) = / sin ms . </> (m) dm, J a J a ^ (m) being a suitable arbitrary function. 2. Determine the residual oscillation in the case when F (s) - (c2 -f s2)-1. * Cf. H. Lamb, Dynamical Theory of Sound, p. 19. 44 The Classical Equations 3. If F (s) = se~h I s I , where h > 0 and # is chosen to be a solution of the differential equation and the supplementary conditions x — 0, x — 0, when t = — oo , x and x continuous at t — 0, the residual oscillation is given by 4th cos kt X^~(k*~+h*)2' If k2 > h2 there is a negative value of t for which r — 0, but if k2 < h2 there is no such value. 4. If .c + k*x = x<*(t)> where o>(/) is zero when t~ oo and is bounded for other real values of t, the solution for which x and b are initially zero can be regarded as the residual oscillation of a simple pendulum disturbed by a transient force. 5. If 0 < a2 < A (t) < 62, the differential equation .- + A(t)x = T) (C) is satisfied only by oscillating functions. Prove that the interval between two consecutive roots of the equation x — 0 lies between rr/a and TT/&. [Let y be a solution o£y 4- 62y = 0 which is positive in the interval r± < t < r2, then Let us now suppose that it is possible for x to be of one sign (positive, say) in the interval T, ^ t < T2 and zero at the ends of the interval. We are then led to a contradiction because the suppositions make .r positive near ^ and negative near r2, they thus make the left-hand side negative (or zero) and the right-hand side positive. Hence the interval between two consecutive roots of x must be greater than any range in which y is positive, that is, greater than y/b. In a similar way it may be shown that if z is a solution of z -f a2z = 0 the interval between two consecutive roots of the equation z — 0 is greater than any range in which x ^ 0.] This is one of the many interesting theorems relating to the oscillating functions which satisfy an equation of type (C). For further developments the reader is referred to Bocher's book, Lemons sur les methodes de Sturm, Gauthier-Villars, Paris (1917) and his article, " Boundary problems in one dimension," Fifth International Congress of Mathematicians, Proceedings, vol. I, p. 163, Cambridge (1912). 6. Prove that x2 -f x2 remains bounded as t -> x but may not have a definite limiting value, x being any solution of (C). [M. Fatou, Compte* Rendus, t. 189, p. 967 (1929).] The generality of this result has recently been questioned. See Note I, Appendix. § 1-43. Motion with a resistance proportional to the velocity. Let us first of all discuss the motion of a raindrop or solid particle which falls so slowly that the resistance to its motion through the air varies as the first power of the velocity. This is called Stokes' law of resistance; it will be given in a precise form in the section dealing with the motion of a sphere through a viscous fluid ; for the present we shall use simply an unknown constant coefficient k and shall write the equation of motion in the form mdv/dt = m'g — kv, (A) where m is the apparent mass of the body when it moves in air, m' is the reduced mass when the buoyancy of the air is taken into consideration and g is the acceleration of gravity. The solution of this equation is _kt kv = m'g + Be m, Resisted Motion 45 where -B is a constant depending on the initial conditions. If v = 0 when t = 0 we have tct kv = m'g(\ - e '*). To find the distance the body must fall to acquire a specified velocity v we write v = x , , , , 7 mvdv/dx = mg — kv, / i 'gr log -- y & Vra 0 - kv If T7 denote the terminal velocity the equation may be written in the form v m'gx = ra F2 log -=%- -- — m Vv. Let us now consider the case in which the particle moves in a fluctuating vertical current of air. Let /' (t) be the upward velocity of the air at time t, v the velocity of the particle relative to the ground and u = v + /x (t) the relative velocity. On the supposition that the resistance is proportional to the relative velocity the equation of motion is mdv/dt = m'g — ku --- m'g — kv ~ kf (t). If v = 0 when t = 0 the solution is The last integral may be written in the form /'(t-r)e "'dr, / 0 which is very useful for a study of its behaviour when t is very large. The distance traversed in time / is given by the equation the constant in/ (t) being chosen so that/ (0) = 0. In particular if ~~~k~ ' p' i ki clc — we have v = — «—- 0- — ,-* [ke m — mP sin pt — k cos pi] , m2p2 + k2L * r r j» ck * — r 7 ^ vn~\ As t -> oo we are left with a simple harmonic oscillation which is not in the same phase as the air current. It should be emphasised that this law of resistance is of very limited application as there is only a small range of velocities and radius of particle 46 The Classical Equations for which Stokes' law is applicable. It should be mentioned that the product of the radius and the velocity must have a value lying in a certain range if the law is to be valid. An equation similar to (A) may be used to describe the course of a unimolecular chemical reaction in which only one substance is being trans- formed. If the initial concentration of the substance is a and at time t altogether x gram-molecules of the substance have been transformed, the concentration is then a — x and the law of mass action gives dx . the coefficient k being the rate of transformation of unit mass of the substance. Simultaneous equations involving only the first derivatives of the variables and linear combinations of these variables occur in the theory of consecutive unimolecular chemical reactions. If at the end of time t the concentrations of the substances A, B and C are x, y and z respectively and the reactions are represented symbolically by the equations A-+B, B-+C, the equations governing the reactions are dx/dt = — &!#, dy/dt = ^x — k2y, dz/dt = k2y. A more general system of linear equations of this type occurs in the theory of radio-active transformations. Let P0, Ply ... Pn represent the amounts of the substances A0, A1 , ... An present at time t, then the law of mass action gives ,p _°- A P dt ~~~ ** OJ ~^ — ^n-lPn-l where the coefficients As are constants. In my book on differential equations this system of equations is golved by the method of integrating factors. This method is elementary but there is another method* which, though more recondite, is more convenient to use. Let us write ps (x) = f V*'P8 (t) dt, (B) J Q Too fjp then e-*« ~8 dt = - P8 (0) + xps (x), Jo dt * H. Bateman, Proc. Camb. Phil. Soc. vol. xv, p. 423 (1910). Reduction to Algebraic Equations 47 and so the system of differential equations gives rise to the system of linear algebraic equations xp, (x) - P0 (0) = - A^ (x), xpl (x) - Pl (0) = AO^O (x) ~ \pl (x), xp2 (x) - P2 (0) = Aj^ (x) - X2p2 (x), Xpn (X) - Pn (0) = V,1>«-1 (X) - from which the functions pQ (x), pt (x), ... pn (x) may at once be derived. If Pl (0) = P2 (0) = ... = Ptt (0) = 0, i.e. if there is only one substance initially, and if P0 (0) = Q, we have p» (x) = x TV Pl (x} = ~(*TAo)7aTXj ' To derive Ps (t) from j9s (x) we simply express ps (x) in partial fractions p te) = - C° ____ h ... - - X ~T" AQ X ~\- Ag The corresponding function Ps (t) is then given by for this is evidently of the correct form and the solution of the system of differential equations is unique. The uniqueness of the function Ps (t) corresponding to a given function ps (x) can also be inferred from Lerch's theorem which will be proved in § 6-29. If some of the quantities Aw are equal there may be terms of type in the representation of ps (x) in partial fractions. In this case the corre- sponding term in Ps (t) is n _A t ^m,* t < e "t . Such a case arises in the discussion of a system of linear differential equations occurring in the theory of probability*. § 1-44. The equation of damped vibrations. A mechanical system with one degree of freedom may be represented at time t by a single point P which moves along the x-axis and has a position specified at this instant by the co-ordinate x. This point P, which may be called the image of the system, may in some cases be a special point of the system, provided that the path of such a point is to a sufficient approximation rectilinear. The * H. Bateman, Differential Equations, p. 45. 48 The Classical Equations mechanical system may also in special cases be just one part of a larger system ; it may, for instance, be one element of a string or vibrating body on which attention is focussed. To obtain a simple picture of our system and to fix ideas we shall suppose that P is the centre of mass of a pendulum which swings in a resisting medium. The motion of the point P is then similar to that of a particle acted upon by forces which depend in value on t, x, x and possibly higher derivatives. For simplicity we shall consider the case in which the force F is a linear function of x and x, ™ /. m , m 07 u. . F = / (t) — h (t) x — 2k (t) x. In the case when h (t) and k (t) are constants the equation of motion takes the simple form rt7 . 9 » /jt. , A v * x + 2kx + n2x = f (t). (A) The motion of the particle is in this case retarded by a frictional force proportional to the velocity. If it were not for this resistance the free motion of the particle would be a simple harmonic vibration of frequency ft/277. The effect of the resistance when n2 > k2 is to reduce the free motion to a damped oscillation of type x = Ae~kt sin (pt + e), (B) where A and € are arbitrary constants and pz = n2 - k2. The period of this damped oscillation may be defined as the interval between successive instants at which x is a maximum and is %TT/P. One effect of the resistance, then, is to lengthen the period of free oscillations. It may be noted that the interval between successive instants at which x = 0 is 2-rr/p. The time range 0 < t < oo may, then, be divided up into intervals of this length. The sign of x changes as t passes from one interval to the next and so the point P does in fact oscillate. Points P and P' of two intervals in which x has the same sign may be said to correspond if their associated times t, t', are connected by the relation pt' = pt + 2m7r, where m is an integer. We then have x' = xe~k(t'~t} = xe~2kmirlp. The positive constant k is seen, then, to determine the rate of decay of the oscillations. When n2 = k2 the free motion is given by x = (A + Bt) e~kt, where A and J5 are arbitrary constants. In this case x vanishes at a time t given by B = k (A -f- Hi), thus | x \ increases to a maximum value and then decreases rapidly to zero. The motion of a dead-beat galvanometer needle may be represented by an equation of type (A) with n2 = k2. Damped Vibrations 49 When k2 > n2 the free motion is of type x = Ae~ut -f fie-"', where ^ and v are the roots of the equation 22 ~ 2A'Z + 7l2 - 0. In this ease the general value of x is obtained by the addition of two terms each of which represents a simple subsidence, the logarithmic de- crement of which is u for the first and v for the second. The time r which is needed for the value of x in one of these subsidences to fall to half value is given by the equation ^ __ 2 — e UT- In the case of the damped oscillation (B) the quantity Ae~ki can be regarded as the amplitude at time t. In an interval of time r this diminishes in the ratio r: 1, where r = ekr. Putting T = 1, we have k -- log r, whence the name logarithmic decre- ment usually given to k. Instead of considering the logarithmic decrement per unit time we may, in the case of a damped vibration, consider the logarithmic decrement per period or per half-period*. It is knjp for the half -period. When / (t) — C sin mt, where C and m are constants, the solution of the differential equation (A) is composed of a particular integral of type ~ (n2 — m2) sin nit — 2km cos mt X ~ C . 0 0\0~ , A 1 9 "«T" ...... (I) (n* — m2)2 -f 4:k~m2 v ' and a complementary function of type (B). The particular integral is obtained most conveniently by the symbolical method in which we write D = djdt and make use of the fact that D2f — — m2/. The operator D is treated as an algebraic quantity in some of the steps ____ -f- 2A-/> H- n2 ~ ^2 - m2 -f 2k 2-m2-- 2kD _ __ 2^ ( '' If .r = 0, a; = 0 when ^ = a the unknown constants in the complementary function may be determined and we find that - r)f(r)dr. ...... (C) This result may be obtained directly from the differential equation by using the integrating factors ekt sin pt and eu cos pt. We thus obtain the equations px (*< sin pt + (x + kx) «*' cos p< = | e*r cos pr ./ (r) dr> cos fc-r) ^' sin p< = ekr sin pr ./ (T) * E. H. Barton and E. M. Browning, Phil. Mag. (6), vol. XLVII, p. 495 (1924). B 4 50 The Classical Equations from which (C) is immediately derived. We also obtain the formula X + kx - [ e~k(t-T)COSp (t - r)f(r) dr. introducing an angle c defined by the equation 2km tan e=2 2 n2 — m2 the particular integral (I) may be expressed in the form x ~ A siri (mi — e), where the amplitude A is given by the formula A2 [(n2 ~ m2)2 + 4k2m2] - C2. N This is the forced oscillation which remains behind when the time t is so large that the free oscillations have died down. The amplitude A is a maximum when m is such that m2 ^ n2 — 2k2 =- p2 — k2. We then have ^4max= C/2kp. Writing A = a/lmax it is easily seen that when m is nearly equal to n and k/n is so small that its square can be neglected we have the approxi- mate formula* 7 , , ,, „. i k = a I n — m \ (1 — a2) *. This formula has been used to determine the damping of forced oscillations of a steel piano wire. It should be noticed that a differential equation of type (A) may be obtained from the pair of equations u -f ku -f nv — 0, v -f kp — nu = 0, where k and n are constants. These represent the equations of horizontal motion of a particle under the influence of the deflecting force of the earth's rotation and a frictional force proportional to the velocity. These equations k uu -f- vv -f k (u2 -f v 2) = 0, u2 + v2 = q2e-2kt, hence k is the logarithmic decrement for the velocity. The equation of damped vibrations has some interesting applications in seismology and in fact in any experimental work in which the motion of the arms of a balance is recorded mechanically. The motion of a horizontal or vertical seismograph subjected to dis- placements of the ground in a given direction, say x — f (t), can be repre- sented by an equation of form 6 + 2kti + n20 + x/l = 0, * Florence M. Chambers, Phil Mag. (6), vol. XLVIII, p. 636 (1924). Instrumental Records 51 where 6 is the deviation of the instrument, k a constant which depends upon the type of damping and I the reduced pendulum length*. The motion of a dead-beat galvanometer, coupled with the seismograph, is governed by an equation of type $ + 2m(j> -f mty + h9 = 0, where </> is the angle of deviation of the galvanometer and h and m are constants of the instrument. To get rid as soon as possible of the natural oscillations of the pendulum, introduced by the initial circumstances, it is advisable to augment the damping of the instrument, driving it if possible to the limit of a periodicity and making it dead beat. By doing this a more truthful record of the move- ment of the ground is obtained. When n2 = k2 the solution of the equation of forced motion x -f 2kx + k*x - / (t) is a= * (t- T)e-k(i-r)f(T)dr^ (A + Bt)e~u. When t is large the second term is negligible and the lower limit of the integral may, to a close approximation, be replaced by 0, — oo, or any other instant from which the value of/ (t) is known. When k is large the second term is negligible even when t has moderate values and if, for such values of t, f (t) is represented over a certain range with considerable accuracy by C sin mt, the value of x is given approxi- mately by the formula f(t) _ Csinmt _ X ~ D* + 2kJ) + T2 " k2 - m2 + 2kD ...(I') When k is large in comparison with m a good approximation is given by k*x =- C sin mt =/(/)> and the factor of proportionality k2 is independent of m, consequently, if a number of terms were required to give a good representation of / (t) within a desired range of values of t, the record of the instrument would still give a faithful representation, on a certain definite scale, of the variation of the force. When k and an are of the same order of magnitude this is no longer true, consequently, if the "high harmonics" occur to a marked degree in the representation of/ (t) by a series of sine functions, the record of the instru- ment may not be a true picture of the forcef. * B. Galitzm, "The principles of instrumental seismology," Fifth International Congress of Mathematicians, Proceedings, vol. I, p. 109 (Cambridge, 1912). f If m = &/10 the solution of the differential equation is approximately k2x — -99 sin (mt - c), where c ia the circular measure of an angle of about 16° 59'. When m = k/5 the solution is approxi- mately kzx — «96 sin (mt — c), where e is the circular measure of an angle of about 31° 47'. 52 The Classical Equations When n ^ k the formula (I') shows that if k is large in comparison with both m and n the solution is given approximately by the formula 2kmx = ~ C cos mt which may be written in the form This result may be obtained directly from the differential equation by neglecting the terms x and n2x in comparison with 2kx. In this case the velocity x gives a faithful record of the force on a certain scale. Finally, if n2 is large in comparison with k and ra, formula (I) gives and the instrument gives a faithful record of the force when the natural vibrations have died down. § 1-45. The dissipation function. The equation of damped vibrations mx + kx + {JLX = f (t) may be written in the form of a Lagrangian equation of motion d dT\ dF 3V where T - \mx\ F - \kx\ V - \^x\ Regarding m as the mass of a particle whose displacement at time t is x, T may be regarded as the kinetic energy, V as the potential energy and F as the dissipation function introduced by the late Lord Rayleigh*. The. function F is defined for a system containing a number of particles by an equation of type p = ^ (/^2 + ^, + ^ where kx, ky, kz are the coefficients of friction, parallel to the axes, for the particle x,y,z. Transforming to general co-ordinates #1 , </2 > <7a > . . . qn we may write where the coefficients [rs], (rs), {rs} are of such a nature that the quadratic forms T, F, V are essentially positive, or rather, never negative. These coefficients are generally functions of the co-ordinates ql9 ... qn , but if we are interested only in small oscillations we may regard ql , ... qn , ql9 ... qn as small quantities and in the expansions of the coefficients in ascending powers of ql9 ... qn it will be necessary only to retain the constant terms if we agree to neglect terms of the third and higher orders in (ft, ... qn, 4i> ••• ?«• * Proc. London Math..Soc. (1), vol. iv, p. 357 (1873). Reciprocal Relations 53 The generalised Lagrangian equations of motion are now - dt\dq where Qm is the generalised force associated with the co-ordinate qm. Since T is supposed to be approximately independent of the quantities qlt q2, ... qn the second term may be omitted. Using rs as an abbreviation for the quadratic operator the equations of motion assume the linear form H& + T2gr2+ ... = Q19 2l?1+ 22g2+ ... = Q2. ...... (E) Since [rs] = [sr], (rs) = (sr), (rs} = {sr}, it follows that 7s = sr. § 1-48. Rayleigh's reciprocal theorem. Let a periodic force Qs equal to As cos pt act on our mechanical system and produce a forced vibration of = KA8 cos (pi - e), where X is the coefficient of amplitude and e the retardation of phase. The reciprocal theorem asserts that if the system be acted on by the force Qr = A cos pi, the corresponding forced vibration for the co-ordinate </s, will be v . . qs = KAr cos (pt — €). Let D denote the determinant IT 12 13 .. 21 22 23 .. 31 32 33 .. and let rs denote its partial derivative with respect to the constituent rs when no recognition is made of the relation rs = sr and when all the constituents are treated as algebraic quantities. This means that rs is the cofactor of rs in the determinant operator D. Solving the equations (E) like a set of linear algebraic equations on the assumption that D ^ 0, we obtain the relations f ...nlQn, ...n2Qn, From a property of determinants we may conclude that since rs = sr we have also rs = sr . Thus the component displacement qr due to a force <?. is given by - 54 The Classical Equations Similarly, the component displacement qs due to a force Qr is given by Dq8 = srQr. Distinguishing the second case by adash affixed to the various quantities, where the coefficients As, A/ may without loss of generality be supposed to be real. If they were complex but had a real ratio they could be made real by changing the initial time from which t is measured. Expressing the solution in the form T W S7* 0 — A "5 pu,t n ' — A Clpt (Jr — ^1 8 j~j V , ys — sir jpG , and defining the forced vibration as the particular integral obtained by replacing d/dt in each of the operators by ip, we obtain the relation A/qr = Asq; which gives reciprocal relations for both amplitude and phase. In the statical case the quantities [rs], (rs), are all zero and D, rs, rs are simply constants. Rayleigh then gives two additional theorems corresponding to those already considered in § 2. (2) Suppose that only two forces Ql3 Q2 act, then If ql = 0 we have ^ ^ 2 = [122- From this we conclude that if q2 is given an assigned value a it requires the same force to keep q^ = 0 as would be required if the force Q2 is to keep q2 = 0 when ql has the assigned value a. (3) Suppose, first, that Q1 ==• 0, then the equations (F) give ?1:?8=21:22. Secondly, suppose q2 = 0, then 02:0!= -12:22. Thus, when Q2 acts alone, the ratio of the displacements ql9 q2 is — Q^IQu where Q11 Q% are the forces necessary to keep q2 — 0. § 1-47. Fundamental equations of electric circuit theory. A system of equations analogous to the system (E) occurs in the theory of electric circuits. This theory may be based on KirchhofTs laws. (1) The total impressed electromotive force (E.M.P.) taken around any closed circuit in a network is equal to the drop of electric potential ex- pressed as the sum of three parts due respectively to resistance, induction and capacity. Electric Circuits 55 Thus, if we consider an elementary circuit consisting of a resistance element of resistance jR, an inductance element of self-induction (or in- ductance) L and a capacity element of capacity (capacitance) (7, all in series, and suppose that an E.M.F. of amount E is applied to the circuit, Kirchhoff's first law states that at any instant of time RI + LTt + 9C = E> W where / is the current in the circuit and Q = \Idt. The fall of potential due to resistance is in fact represented by KI where R is the resistance of the circuit (Ohm's law), the drop due to in- ductance is Ldl/dt and the drop across the condenser is Q/C. There is an associated energy equation in which RI represents the rate at which electrical energy is being converted into heat, while the second and third terms represent rates of increase of magnetic energy and electrical energy respectively. The right-hand side represents the rate at which the impressed E.M.F. is delivering energy to the circuit, while the left-hand side is the rate at which energy is being absorbed by the circuit. The inductance element and the condenser may be regarded as devices for storing energy, while the resistance is responsible for a dissipation of energy since the energy converted into Joule's heat is eventually lost by conduction and radiation of heat or by conduction and convection if the circuit is in a moving medium. If we regard Q as a generalised co-ordinate, we may obtain the equation (I) by writing y = ^ f _ ^ V = Q2/2C> dF dV _, . - . f .- = E. dQ vQ (2) In the case of a network the sum of the currents entering any branch point in the network is always zero. If we consider a general form of network possessing n independent circuits, Kirchhoff's second law leads to the system of equations -Er, (II) (r= 1,2, ...n), where Er is the E.M.F. applied to the rth circuit, Lss, R9SJ CKS denote the total inductance, resistance and capacitance in series in the circuit s, while Lrs, Rrs, Crs denote the corresponding mutual elements between circuits r and s. We have written Frs for the reciprocal of Crs and Qs for I/, eft, where Is is the current in the 5th circuit or mesh. 56 The Classical Equations An elaborate study of these equations in connection with the modern applications to electrotechnics has been made by J. R. Carson*. In discussing complicated systems of resistances, inductances and capacities it will be convenient to use the symbol (LRC) for an inductance L, a resistance R and a capacity C in series. If a transmission line running between two terminals T and T' divides into two branches, one of which contains (LRC) and the other (L'R'C'}, and if the two branches subse- quently reunite before the terminal T' is reached, the arrangement will be represented symbolically by the scheme In electrotechnics a mechanical system with a period constant n and a damping constant k is frequently used as the medium between the quantity to be studied (which actuates the oscillograph) and the record. The oscillograph is usually critically damped (k = n) so as to give a faithful record over a limited range of frequencies but even then the usefulness of the instrument is very limited as the range for accurate results is given roughly by the inequality 10m < n. A method of increasing the working range of such an oscillograph has been devised recently by Wynn- Williams f. If an E.M.F. of amount E acts between two terminals T and T' and the aim is to determine the variation of E, the usual plan is to place the oscillograph 0 in line with T and T' so that T, 0 and T' are in series. In our notation the arrangement is T - 0 - T. Instead of this Wynn- Williams proposes the following scheme in which L, R and C are chosen so that 2k = JR/L, n2 = l/CL and L19 K19 C1 are chosen so that Ll, C\ are small and R1 is such that there is a relation (k 4- ^)2 - n2 + n^9 where 2k, - J?,//,, nf = l/LC^ Putting L) = 0 and writing q for the current flowing between T and T' , x for the reading of the oscillograph, F for the back E.M.F. of the system (LRC), we have Therefore x = -~; c\ [D2 + 2kD + n2] x = F. E * Klcctnc circuit theory and operational calculus (McGraw Hill, 1926). f Phil. Mag. vol. L, p. 1 (1925). Cauchy's Method 57 Hence when the reading of the oscillograph is used to determine E the oscillograph behaves as if its period constant were (n2 -f n^ instead of n and its damping constant k -f 1cl instead of k. By choosing n^ = 24n2, &x = 4k = 4n we have N = (n2 + ft!2)* = 5n, Ar - k + ^ - 5fc - 5w, thus we obtain an amplitude scale which is the same as that of an oscillo- graph with a period constant 5n and a damping constant 5 A1. § 1-48. Cauchy^s method of solving a linear equation*. Let us suppose that we need a particular solution of the linear differential equation <f>(D)u = f(t), ...... (I) where (f> (D) = a0i> + ^D"-1 4- ... an, and D denotes the operator djdt. The coefficients #s are either constants or functions of t. For convenience we shall write a (t) = l/a0. If (v1} v2, ... #n) are distinct solutions of the homogeneous equation the coefficients Cl , C2 , . . . Cn in the general solution t> = ^1^1 -f ^2^2+ ••• GnVn may be chosen so that v satisfies the initial conditions v (r) = v' (r) = ... ^^~2> (r) - 0, v^-1* (T) = a (T)/ (T), where i;(s) (£) denotes the ,sth derivative of v (t) and r is the initial value of t. We denote this solution by the symbol v (t, r) and consider the integral u (t) - [ v (t, r) dr. Jo Assuming that the differentiations under the integral sign can be made by the rule of Leibnitz, we have rt =\ Jo = 1,2, ...n- 2,n- the terms arising from the upper limit vanishing on account of the pro- perties of the function v. On the other hand Dnu = D"v (t, r)dr-\-a (t)f(t), Jo and so <f> (D) u = / (t) + f V (D) v . dr = / (t). Jo This particular solution is characterised by the properties u (0) - u' (0) = ... tt<"-i> (0) - 0, w(n) (0) = a (O)/ (0). If, when / (0 = 1> ^ (^ T) == e/r (£, r), the general value for an arbitrary * Except for some slight modifications this presentation follows that of F. D. Murnaghan, Bull. Amer. Math. Soc. vol. xxxm, p. 81 (1927). 58 The Classical Equations function / (t) is seen to be / (t) if/ (t, T) and so we may write u (t) in the form t u(t)=\ t(t,T)f(t)dr. ...... (II) Jo Introducing the notation = J we have -g— £ (t, r) = - iff (t, T), and the integral in (II) may be integrated by parts giving u (t) =/(0) f (t, 0) + f (*, r)/' (r) dr. Jo This is the mathematical statement of the Boltzmann-Hopkinson principle of superposition, according to which we are able to build up a particular solution of equation (I) from a corresponding particular solution £ (t, r) for the case in which /W=l t>*> = 0 t< T. When the coefficients in the polynomial </> (D) are all constants we may Write a0 (D - rt) (D - r,) . . . (Z> - rj, where r1? r2, ... rn are the roots of the algebraic equation <f> (x) = 0. Taking first the case in which these roots are all distinct, we write vs = er»«-T> s = 1, 2, ... n. The equations to determine the constants C are then We may solve for C by multiplying these equations respectively by the coefficients of the successive powers of x in the expansion (/y* — — V \ ( *Y — V \ If , ff \ • — /j - 1 , r) <•>» I h /j»W — 1 x — r2) (x — r3) ... (x — rn) — o0 -t- u^x -t- ... vn-iX Since bn_t = 1 we find that «/ (^ = Ci (T! - r2) (r± - r3) ... fa - rn) = aCtf fa), and the other coefficients may be determined in a similar way. Writing kr for the reciprocal of <£' (rs) we have 00 v (f -\ _ f /-\ V ^ ^^(i-r) _ /" /-.\ ./. // -\ v \l'> T/ — y VT/ ^ ^se — 7 \T/ T \^y T)> 8-1 and it ps denotes the reciprocal of r8 n g (t, r) = 2 paka [ers(i~r) — 1]. a-l Heaviside's Expansion 59 i n if and so [<£ (O)]-1 - - S p^, When there is a double root rl = r2, we write ^ = eri«-T>, v2 - (t - T) efi«-T), v3 , ... vn being the same as before. The equations to determine the constants Carenow <74 + 0 . <7, + C7, + ... 0. = 0, r^ + 1 . <72 + r3 . (73 + ... rn . Cn = 0, /y-i/7, + (»-!) r.-^C, + rj-'C1, + ... r,,--^, = af (t). Writing (r - A) F (r) = (r - A) (r - r3) ... (r - rn) = c0 + ctr + ... c^^"-1 and multiplying the equations by c0, clt ... cn_t respectively we find that, since Cn_r = 1, G1 (r, -\)F (r,) + Cz [F (r,) + (r, - A) F' (r,)] = af (r). The quantity A is at our disposal. Let us first write A = r1} we then have /-i n / \ f i •, CtF (rj = af(r). Simplifying the preceding equation with the aid of this relation we °btain C,F (r,) + C2F' (r,) = 0. Writing G (r) = ajF (r), we have C, = / (T) G' (r,), Ct =f (T) O (r,). These are just the constants obtained by writing <f>(r) r-fj (r-rtf r - r3 ">-rn' and a similar rule holds in the case of a multiple root of any order or any number of multiple roots. Thus in the case of a triple root, § 1-49. Heaviside's expansion. The system of differential equations (II) of § 1-47 may be written in the form 2 af.C.=X(0, where the a's are analogous to the operators ~rs of § 1-45. 60 The Classical Equations Denoting the determinant | ar<s \ by <f> (D) and using Ars to denote the co-factor of the constituent ars in this determinant, we have <£ (D) Q, = 2 ArsEr (t). r=l To obtain an expansion for Qs we first solve the equation j (D) ys (t) = 1 with the supplementary conditions y,(0) = y/(0) = ...y.<«»-»(0)=o, then xrs = Asry3 (t) is a particular solution of the equation </> (D) xr — Asr . 1 for which a:r(0) = *r'(0) = 0; and by the expansion theorem of § 1-48, - xrs (t, 0) = Asrys (t, 0) = s A_.(!-Ler..+ ^-0> jT,(ra)^(ra)e + .£(0) ' which is Heaviside's expansion formula. The corresponding formula for <UOi- & (0 = S #r (0) *ir (*, 0) + tf/ (r) *sr (t, r) dr , r-l L JO J and this particular solution satisfies the conditions Qs (0) = Q.' (0) = o. § 1-51. The simple wave-equation. There are a few partial differential equations which occur so frequently in physical problems that they may be called classical. The first of these is the simple wave-equation which occurs in the theory of a vibrating string and also in the theory of the propagation of plane waves which travel without change of form. These waves may be waves of sound, elastic waves of various kinds, waves of light, electromagnetic waves and waves on the surface of water. In each case the constant c represents the velocity of propagation of a phase of a disturbance. The meaning of phase may be made clear by considering the particular solution V = sin (x — ct) which shows that F has a constant value whenever the angle x — ct has a constant value. This angle may be called the phase angle, it is constant for a moving point whose x-co-ordinate is given by an equation of type x = ct -I- a, where a is a constant. This point moves in one direction with uniform velocity c. There is also a second particular solution V = sin (x -\- ct) Wave Propagation 61 for which the phase angle x -f ct is constant for a point which moves with velocity c in the direction for which x decreases. These solutions may be generalised by multiplying the argument x ± ct by a frequency factor 27TV/C, where v is a constant called the frequency, by adding a constant y to the new phase angle and by multiplying the sine by a factor A to represent the amplitude of a travelling disturbance. In this way the particular solution is made more useful from a physical standpoint be- cause it involves more quantities which may be physically measurable. In some cases these quantities may be more or less determined by the supple- mentary conditions which go with the equation when it is derived from physical principles or hypotheses. Usually this particular equation is derived by the elimination of the quantity U from two equations dv du zu _ dv ... ~di~adx' ~dt~pfa ...... IA) involving the quantities U and F, the coefficients a and ft being constants. The constant c is now given by the equation c2 - aft. It should be noticed that if F is eliminated instead of 17, the equation obtained for U is and is of the same type as that obtained for V. This seems to be a general rule when the original equations are linear homogeneous equations of the first order with constant coefficients, however many equations there may be. The rule breaks down, however, when the coefficients are functions of the independent variables. If, for instance, a and ft are functions of x the resulting equations are respectively 327 a cw . a / du\ dt2 dx \ dx / ' These equations may be called associated equations. Partial differential equations of this type occur in many physical problems. If, for instance, y denotes the horizontal deflection of a hanging chain which is performing small oscillations in a transverse direction, the equation of vibration is 82y __ a / dy\ where g is the acceleration of gravity and x is the vertical distance above the free end. Equations of the above type occur also in the theory of the propagation of shearing waves in a medium stratified in horizontal plane layers, the physical properties of the medium varying with the depth. 62 The Classical Equations § 1-52. The differential equation (I) was solved by d'Alembert who showed that the solution can be expressed in the form V=f(x-ct) + g(x + ct), where / (z) and g (z) are arbitrary functions of z with second derivatives /" (z)> $" (z) that are continuous for some range of the real variable z. A solution of type f (x — ct) will be called a " primary solution," a term which will be extended in § 1-92 to certain other partial differential equations. To illustrate the way in which primary solutions can be used to solve a physical problem we consider the transverse vibrations of a fine string or the shearing vibration of a building*. The co-ordinate x is supposed to be in the direction of the undisturbed string and in the vertical direction for the building, the co-ordinate y is taken to represent the transverse displacement. If A denotes the area of the cross-section, which is a horizontal section in the case of the building, and p the density of the material, the momentum of the slice Adx is M dx, where M — pA ^ . The slice is acted upon by two shearing forces acting in a transverse direction and by other forces acting in a "vertical" direction, i.e. in the diiection of the undisturbed string. Denoting the shearing force on the section x by S, that on the section x -f dx is S -f ~ dx. dS dx The difference is x dx, and so the equation of motion is ox m __ ss St ~dx' We now adopt the hypothesis that when the displacement y is very small ~ where /JL is a constant which represents the rigidity of the material in the case of the building and the tension in the case of the string. According to this hypothesis if p and A are also constants where c2 = p/p. The expressions for M and S also give the equation as BM p-ft-P-to' and so we have two equations of the first order connecting the quantities M and 8', these equations imply that M and S satisfy the same partial differential equation as y. i * The shearing vibrations of a building have been discussed by K. Suyehiro, Journal of the Institute of Japanese Architects, July (1926). Transmission of Vibrations 63 In the case of the building one of the boundary conditions is that there is no shearing force at the top of the building, therefore 8 = 0 when x = h. Assuming that y=f(x-ct) + g(x + ct) di/ the condition is ^ = 0 when x = h. and so ox 0 = /' (A - ct) + g' (h + ct). This condition may be satisfied by writing y = <f> (ct + x — h) ~h (f> (ct — x + 7^), where </> (2) is an arbitrary function. A motion of the ground (x = 0) which will give rise to a motion of this kind is obtained by putting x = 0 in the above equation. Denoting the motion of the ground by y = F (t) we have the equation F (t) = <f> (ct - h) -\- </> (ct 4- h) ...... (A) for the determination of the function </> (z). In the case of the string the end x = / may be stationary. We therefore put y = 0 for x = I and obtain the equation 0=f(l-ct) + g(l + ct) which is satisfied by where $ is another arbitrary function. If the motion of the end x = 0 is prescribed and is y = 0 (t) we have the equation -0 (t) = </r (Ct - 1) - $ (Ct + I) ...... (B) for the determination of the function $ (z). If, on the other hand, the initial displacement and velocity are pre- scribed, say ~ y=e(x), gf = *(«) when t = 0, we*have the equations 0(x)=f (x) + g (x), x (x) = c [<?' (x) - /' (x)] which give 2cf (x) = c9' (x) - x(x)> 2cg' (x) = cff (x) + x (x), and the solution takes the form 1 rx+ct y=\\0(x~ct) + e(x + ct)-\+ f\ x(T)dr. L(j J x - ct If in the preceding case both ends of the string are fixed, the equation (B) implies that ^ (x) is a periodic function of period 21, the corresponding time interval being 2l/c. Submultiples of these periods are, of course, admissible, and the inference is that a string with its ends fixed can perform oscillations in which any state of the system is repeated after every time 64 The Classical Equations interval of length 2lm/nc, where m and n are integers, n being a constant for this type of oscillation. In the case of the building the ground can remain fixed in cases when (f> (z) is a periodic function of period 4=h such that <f) (z -f 2h) = — <f) (z). It should be noticed that the conditions of periodicity may be satisfied by writing 0 (z) = sin - j- , (f> (z) = sin [(n + |) TTZ/&], where in and n are integers. Thus in the case of the string with fixed ends there are possible vibrations of type mirx irnrct - cos y-~, and in the case of the building with a free top and fixed base there are possible vibrations of type . . r/ iv7^"! r/ ivfrttfi y = bm sin \(n + J) , | cos \(n -f J) -,- . L ^ J L ^ J These motions may be generalised by writing for the case of the string « . mnx mnct y - S am sin - cos — y— , m-l * * where the coefficients aw are arbitrary constants. For complete generality we must make s infinite, but for the present we shall treat it as a finite constant. The total kinetic energy of the string is , [i . mnx . HTTX -, since we have sm 7 sm — . ax = 0 n + m Jo * * = Z/2 ft = m. Since the kinetic energy is the sum of the kinetic energies of the motions corresponding to the individual terms of the series, these terms are supposed to represent independent natural vibrations of the string. These are generally called the normal vibrations. The solution for the vibrating building can also be generalised so as to give 3 y= S bn sin \(n -f J) \- cos \ (n -f i) -,- L • n-i L ^J L h \ and the kinetic energy is in this case f, (271-f 1)27T2C2, 2 . > ^ ' *x 2 nt v^ Vibration of a String 65 for now we have corresponding relations / sin \(n 4- J) ^- sin (m -f J) ^- ute = 0 m±n = A/2 m — n. Such relations are called orthogonal relations. § 1-53. In the case of the equation of the transverse vibrations of a string there is a type of solution which can be regarded as fundamental. Let us suppose that the point x = a is compelled to move with a simple harmonic motion* n , t . y = p cos (pt 4- «), where a, /3 and p are arbitrary real constants. If the ends x = 0, x = I remain fixed, it is easily seen that the differential equation W dx* ' and the conditions y = 0 at the ends may be satisfied by writing y — y± for 0 < x < a and y = y2 for a < x < /, where yl = p cosec (Xa) sin (\x) cos (p£ -fa), 2/2 = /? cosec A (/ — a) sin A (/ — #) cos (pt + a), and Ac = p. The case of a periodic force F = F0 cos (^tf -f a) concentrated on an infinitely short length of the string may be deduced by writing down the condition that the forces on the element must balance, the inertia being negligible. This condition is F = Py/ - PyJ for x = a, where P is the pull of the string. Substituting the values of y^ and y% we get cF0 = pfiP [cot Aa + cot A (I - a)] . F Therefore /? = p^ cosec XI sin Aa sin A (I — a). The solution can now be written in the form F y= j>0 (*>«)> , , . sin A# sin A (Z — a) where <7 (x, a) = - c— ^— ^ -- - 0 < x < a A sin AL _ sin A (I — x) sin Aa 7 ' = ----- x- — . — ^j ----- a ^ x ^ /. A sin XI This function gr (a;, a) is a solution of the differei^tial equation S+*V-0 ...... (A) * Rayleigh, Theory of Sound, vol. I, p. 195 66 The Classical Equations and satisfies the boundary conditions g = 0 when x — 0 and when x — I. It is continuous throughout the interval 0 < x < I, but its first derivative is discontinuous at the point x = a and indeed in such a manner that lim 8->Q The function g (x, a) is called a Green's function for the differential d2u expression , ^ 4- A2w, it possesses the remarkable property of symmetry expressed by the relation g (x, a) = g (a, x). This is a particular case of the general reciprocal theorem proved by Maxwell and the late Lord Rayleigh. It should be noticed that the Green's function does not exist when A has a value for which sin XI = 0? that is, a value for which the equation (A) possesses a solution g = sin Xx which satisfies the boundary conditions and is continuous (D, 1) throughout the range (0 < x < /). A fundamental property of the Green's function g (x, a) is obtained by solving the differential equation by the method of integrating factors. Assuming that y is continuous (Z>, 2) in the interval (0, 1) and that the function/ (x) is continuous in this interval, the result is that U = U ° - U a-0 ~ l Jo where u (0) and u (I) are assigned values of u at the ends. If these values are both zero y is expressed simply as a definite integral involving the Green's function and/ (a). § 1-54. The torsional oscillations of a circular rod are very similar in character to the shearing oscillations of a building. Let us consider a straight rod of uniform cross-section, the centroids of the sections by planes x = constant, perpendicular to the length of the rod, being on a straight line which we take as axis of x. Let us assume that the section at distance x from the origin is twisted through an angle 6 relative to the section at the origin. It is on account of the variation of 8 with x that an element of the rod must be regarded as strained. The twist per unit length at the place x is defined to be B8 T = ^; it vanishes when 8 is constant throughout the element bounded by the Torsional Oscillations 67 planes x and x -f dx, i.e. when this element is simply in a displaced position just as if it had been rotated like a rigid body. The torque which is transmitted from element to element across the plane x is assumed to be Kfir, where /A is an elastic constant for the material (the modulus of rigidity) and K is a quantity which depends upon the size and shape of the cross-section and has the same dimensions as 7, the moment of inertia of the area about the axis of x. Let p denote the density of the material, then the moment of inertia abou^ the axis of x of the element previously considered is pldx and the angular momentum is plOdx. Equating the rate of change of angular momentum to the difference between the torques transmitted across the plane faces of the element, we obtain the equation of motion . d26 „ cW pi = fih „ cl2 ^ fix* which holds in the case when the rod is entirely free or is acted upon by forces and couples at its ends. In this case the differential equation must be combined with suitable end conditions. A simple case of some interest is that in which the end x — 0 is tightly clamped, whilst the motion of the other end x — a is prescribed. § 1-55. The same differential equation occurs alstfin the theory of the longitudinal vibrations of a bar or of a mass of gas. Consider first the case of a bar or prism whose generators are parallel to the axis of x. Let x -f £ denote the position at time t of that cross- section whose undisturbed position is x, then £ denotes the displacement of this cross-section. An element of length, 8x, is then altered to 8 (x -f- f ), of (1 -f £') $x, where the prime denotes differentiation with respect to x. Equating this to (1 + e) $x we shall call e the strain. The strain is thus the ratio of the change in length to the original length of the element and is given by the formula ^ e = ' dx According to Hooke's law stress is proportional to strain for small displacements and strains. The total force acting across the sectional area A in a longitudinal direction is therefore F = EeA, where E is Young's modulus of elasticity for the material of which the rod is composed. The stress across the area is simply Ee. The momentum of the portion included between the two sections with co-ordinates x and x -f 8x is M 8x, where M = pA ~ and p is the density of the material. The equation of motion is then <W W dt ~ dx9 5-2 68 The Classical Equations AS*€ A' When the material is homogeneous and the rod is of uniform section the equation is ^ a^^cte5' where c2 = £//o. Since the modulus E for most materials is about two or three times the modulus of rigidity /z, longitudinal waves travel much more rapidly than shearing waves and the frequency of the fundamental mode of vibration is higher for longitudinal oscillations than it is for shearing oscillations. In the case of a thin rod shearing oscillations would not occur alone but would be combined with bending, and the motion is different. The fundamental frequency for the lateral oscillations is, however, much lower than that for the longitudinal oscillations. Let us next consider the propagation of plane waves of sound in a direction parallel to the axis of x. Let VQ = A 8x be the initial volume of a disc-shaped mass of the gas through which the sound travels, v = A8 (x -f £) the volume of the same mass at time t. We then have v = vQ(l -f e), where e is now the dilatation. If pQ is the original density and p the density of the mass at time t, we may write P = PQ (1 + s), where s is the condensation, it is the ratio of the increment of density to the original density. Since pv = pQvQ we have (1 + s) (1 + e) = 1, and if s and e are both small we may write ~ . ox To obtain the equation of motion we assume that the pressure varies with the density according to some definite law such as the adiabatic law where p0 is the pressure corresponding to the density y and is a constant which is different for different gases. This law holds when there is no sensible transfer of heat between adjacent portions of the gas. Such a state of affairs corresponds closely to the facts, since in the case of vibration of audible frequency the con- densations and rarefactions of our disc-shaped mass of gas follow one another with a frequency of 500 or more per second. For small values of s we may write P = Po(l + 7s)- Waves of Sound 69 The equation of motion is now m _SF St ~ Sx ' where M = PoA ^, F = - Ap. Substituting the values of p and s we obtain the equation a«f_ a«f ~ in which c» = y° Po For sound waves in a tube closed at both ends the boundary conditions are £ =. 0 when x = 0 and when # = I. The solution is just the same as the solution of the problem of transverse vibration of a string with fixed ends. For sound waves in a pipe open at both ends and for the longitudinal vibrations of a bar free at both ends we have the boundary conditions when x = 0 and when x = 19 which express that there is no stress at the ends. The normal modes of vibration are now of type .. ~ nnrx (rmrct\ g=Cm COS -j- COS ^-y- J , where Cm is an arbitrary constant and m is an integer. This solution is of type £ = <l>(x + ct) + <f>(ct- x), and may be interpreted to mean that the progressive waves represented by ^ = <f> (ct — x) are reflected at the end x = 0 with the result that there is a superposed wave represented by £> = <£ (ct -f x). There is a different type of reflection at a closed end of a tube (or fixed end of a rod), as may be seen from the solution £ = ^ (ct - x) - <f> (ct + x), which makes f = 0 when x = 0. Reflection at a boundary between two different fluid media or between two parts of a bar composed of different materials may be treated by introducing the boundary condition that the stress and the velocity must be continuous at the boundary. If progressive waves represented by £0 = #0</> (t — x/c) approach the boundary x = 0 from the negative side and give rise to a reflected wave £x = a^ (t 4- x/c) and a transmitted wave £2 = ^2^ (^ "~ #/c/)> the boundary conditions- are fit fit fit ' " ** ** 70 The Classical Equations where K = yp0 and K = y'p0 , the constants y and y' referring respectively to the media on the negative and positive sides of the origin. The equi- librium pressure pQ is the same for both media. 9& 9fi 9& Now -* = «0, -g^ =-<»i, * = c'*, hence, when x = 0, c$0 - o^' (0, - cs! = <*!</>' (t), c'3ij= a^ (t), and c (s0 — s^) = c's2, K (SQ -\- Sj) — i<'s. mi . — Iherefore sl — — «s0 $ = - — 5 1 jc'c-f KC' ° 2 //C-f ice' °' — KC t , , / • KC+ KC' /c'C+ ACC7 § 1-56. The simple wave-equation occurs also in an approximate theory of long waves travelling along a straight canal, with horizontal bed and parallel vertical sides, the axis of x being parallel to the vertical sides and in the bed (see Lamb's Hydrodynamics, Oh. vm). Let 6 be the breadth of the canal and h the depth of the fluid in an initial state at time t — 0 when the fluid is at rest and its surface horizontal. We shall denote the density of the fluid by p and the pressure at a point (x, y, z) by p. The motion is investigated on the assumption that p is approximately the same as the hydrostatic pressure due to the depth below the free surface. This means that we write P = Po + 9P (h + ri- y), ...... (I) where ^pQ is the external pressure, which is supposed to be uniform, 77 is the elevation of the free surface above its undisturbed position and g is the acceleration of gravity. One consequence of this assumption is that there is no vertical acceleration, in other words, the vertical acceleration is neglected in making this approximation. If, in fact, we consider a small element of fluid bounded by horizontal and vertical planes parallel to the planes of reference, the axis of y being vertically upwards, the equations of motion are pa . Sx8ySz = — ~ 8x . 8y$z, pft . 8x8ySz = — ~ - Sy . 8z8x — pg8x8y8z, d%) py . 8x8y8z = — « 8z . 8x8y, where a, j8, y are the component accelerations. With the above assumption we have 8 = y = 0, and so ^ op Waves in a Canal 71 The assumption of no vertical acceleration is not equivalent to the assumption (I), because an arbitrary function of x, z and t could be added to the right-hand side of (I) and the equations of motion would still give no vertical acceleration. Equation (I) gives pa = — gp x- . This expression for 2 is independent of y, consequently, since g is assumed to be constant, the acceleration a is the same for all particles in a vertical plane perpendicular to the axis of x. The horizontal velocity u depends on x and t only. Now let £ be the total displacement from their initial position of the particles which at time t occupy the vertical plane x. Each particle is supposed to have moved horizontally through a distance £, but actually some of the particles will have moved slightly upwards or downwards as well. the fluid which occupies the region QQ'X'X is supposed to have initially occupied the region PP'A'A. Equating the amount of fluid in the region QQ'N'N to the difference of the amounts in the regions PNXA, P'N'X'A' we obtain the equa- tion of continuity N 'N' A A X X' Fig. 9. or i = - h —- . dx .(II) A second equation is obtained by writing a = >,— . This is approximately ot true in the case of infinitely small motions, the exact equation being a — du Bu u dt^"dx ?=!*&, du Writing we have |^2 = ~~ = a = - g g. (Ill) The equations (II) and (III) now give the wave-equations where c2 = gh. When, in addition to gravity, the fluid is acted upon by small dis- turbing forces with components (X, Y) per unit mass of the fluid, the 72 The Classical Equations assumption that the pressure is approximately equal to the hydrostatic pressure leads to the equation - (g-Y)dy. v TU- • P i vx*7 Thisgives £-/>&- Y^x and the equation of horizontal motion du „ dp '•-fc-^-fc indicates that in general u depends on y as well as on x and t. With, however, the simplifying assumptions that Y is small compared dY with g and that h ~ is small in comparison with X the equation takes the form 3 _ and, if X depends only on x and t, this equation indicates that u is inde- pendent of y. We may then proceed as before and obtain the equations EXAMPLES 1. An elastic bar of length I has masses m0, m1 at the ends x — 0, x — I respectively. Prove that the terminal conditions are Jj] A - _ — vn wVi (*T\ y ' — . 0 ^^ o- ~ mQ 1*9. wiit-ii & — v, Prove that the possible frequencies of vibration are given by the equation (1 - KHiP) tan 9 + U + /*i) ^ = 0, where c2m^ = lAE^t c2ml = lAE^ilt 0 = nl, and nc/27T is the number of vibrations per second. 2. If a prescribed vibration £ = C cos w£ is maintained at the end x = 0 of a straight pipe which is closed at the end x = I the vibration at the place x is given by ~ nl . n (I — x) f = C cosec - sin — cos nt. c c Obtain the corresponding solution for the case in which the end x = I is open. 3. Discuss the longitudinal oscillations of a weighted bar whose upper end is fixed. Systems of Equations 73 4. If ^ and A is an arbitrary constant, the function y — A [sin 2sna — sin satisfies the differential equation _0 _0 d*y _ d*y 2 ~ z ' and the end conditions y — 0 when x = 0 and when x = a + vt. Prove also that when v -> 0, 0 , . SnX Snct it -> 2 A sm — cos — . a a [T. H. Havelock, Phil. Mag. vol. XLVH, p. 754 (1924).] 5. Prove that if y — 0 when x = 0 and x = vt, y =/(*). y = 9 (*)> a solution of ~ ^ = c2 ^~ is given by ~ ^ where a log -- = 2n, exp (r) = ~ - -£- ^K + «) = i/(«) -f I fX g(x)dx, foz Cz«o ^c .' 0 Bn - - - f * f K + *) cos (naa>) -^- - , 7T J -Vt0 MO + 3? and it is supposed that . . . . , % FF^ /(-*) = -/(*), g(-*)--^(«). [E. L. Nicolai, PM. Jfogr. vol. XLIX, p. 171 (1925).] § 1-61. Conjugate functions and systems of partial differential equations. If in equations ((A) § 1-51) we write a = 1, /? = - 1 and use the variable y in place of t we obtain the equations W^dV dU^^dV dx dyy dy ~ ~dx satisfied by two conjugate functions U and V. In this case both functions satisfy the two-dimensional form of Laplace's equation _ dx* 3^2 ~ - This equation is important in hydrodynamics and in electricity and magnetism. The equations (A) may be generalised in another way by writing .97 dU 3V 74 The Classical Equations where a, /?, y, 8, 6, </>, A, /x, a, r are arbitrary constants. In particular, the equations w ^V dU dt ==SK'dx' ~dx^V i A+ ^ f dU lead to the equation -~-- = K ~ 2 , which is the equation for the conduction of heat in one direction when U is interpreted as the temperature and K as the diffusivity. The same equation occurs in the theory of diffusion. It should be noticed that the quantity V satisfies the same equation. Again, if we write -~— = L -x- -f RU, du^cdv + sv dx ° dt ^ ' and interpret V as electric potential, U as electric current, we obtain the differential equation which governs the propagation of an electric current in a cable*. The coefficients have the following meanings : R L C - S resistance inductance capacity leakance all per unit of length of the cable. The quantity U satisfies the same differential equation as V. This differential equation may be reduced to a canonical form by introducing the new dependent variables u, v, defined by the equations TT mlT T7 p./r J ^ u = UeRtiL, v = Vem<L. These variables satisfy the equations dv ,- du and the canonical equations of propagation are Heaviside's equations These equations are of the simple type (I) if SL = CR. In this case a wave can be propagated along the cable without distortion. * Cf . J. A. Fleming, The Propagation of Electric Currents in Telephone and Telegraph Circuits, ch. v. The Telegraphic Equation 75 When dealing with the general equations (A) it is advantageous to use algebraic symbols for the differential operators and to write 3 n d -D • si n" te~Dx' the differential equations may then be written symbolically in the form (0Dt - yDx - /*) F = (aD. + A) U, (<f>Dt -Wx-a)U = (fiDx + r) V. The first equation may be satisfied by writing U=(ODt-yD.-rtW, V=(aDx+\)W, ...... (B) where W is a new dependent variable. Substituting in the second equation we obtain the following equation for W, [(0Dt - yDx - p) (^Dt - 8DX - a) ~ (aDx + A) ($DX +r)]W^ 0, which, when written in full, has the form 9217 9217 921^ /v / f rr / n& , i \ V r" . / & /*>\ U " i f\ - (ar -f j3A + ycr + 8/i) ~ + (AIO- - XT) W = 0. When this equation has been solved the variables U and V may be determined with the aid of equations (B). It is easily seen that U and V satisfy the same equation as W. The equation for W is said to be hyperbolic, parabolic or elliptic according as the roots of the quadratic equation 6</>X* - (68 + fa) X + yS - aft - 0 are real and distinct, equal or imaginary. In this classification the co- efficients a, /?, y, S, By fi, A, /z, a, r are supposed to be all real, the simple wave-equation is then of hyperbolic type, the equation of the conduction of heat of parabolic type and Laplace's equation of elliptic type. The telegraphic equation is generally of hyperbolic type, but if either C = 0 or L = 0 it is of parabolic type and the canonical equation is of the same form as the equation of the conduction of heat. The foregoing analysis requires modification if the coefficients a, j8, y, 8, 0, <f>, A, /A, a, r are functions of x and t, because then the operators aDx -f- A and 9Dt — yDx — \L are not commutative in general, and so the first equation cannot usually be satisfied by means of the substitution (B). If, however, the conditions ^^ da 76 The Classical Equations are satisfied the operators are commutative (permutable) and a differential equation may be obtained for W. In this case the variables U and F do riot necessarily satisfy the same partial differential equation. This is easily seen by considering the simple case when the first equation is U = 0dV/dt and /? and r are independent of t. Differential operators which are not permutable play an interesting part in the new mechanics. § 1-62. For some purposes it is useful to consider the partial difference equations which are analogous to partial differential equations in which we are interested. The notation which is now being used in Germany is the following * : u (x + h, y) - u (x, y) = hux, u (x, y -f h) - u (x, y) = huv, u (x, y} — u (x — h, y) = hu^, u (x, y) — u (x, y — h) = hu^, u (x + h,y) — 2u (x, y) -f u (x — h, y) = Ji2ux^ = h^u^x - The equations ux — Vy> uy — — v% are analogous to those satisfied by conjugate functions since they imply Ux7 + Vvy = 0, Vx* + Vyy = 0. The equations % — vv , u^ — vx give the equations ux2 = u$ , vx^ = vg analogous to the equation of the conduction of heat. § 1-63. The simultaneous equations from which the final partial differential equation is derived need not be always of the first order. In the theory of the transverse vibrations of a thin rod the primary equations where 77 is the lateral displacement,^ the bending moment, A the sectional area, x the radius of gyration of the area of the cross-section about an axis through its centre of gravity, p the density and E the Young's modulus of the material. The resulting equation is of the fourth order. The equation is usually simplified by the omission of the second term. This process of approximation needs to be carefully justified because it will be noticed that the term omitted involves a derivative of the fourth order, that is a derivative of the highest order. Now there is a danger in omitting terms involving derivatives of the highest * See an article by R. Courant, K. Friedrichs and H. Lewy, Math. Ann. Bd. c, S. 32 (1928). f Cf. H. Lamb, Dynamical Theory of Sound, p. 121. Vibration of a Rod 77 order because their coefficients are small. This may be illustrated in a very simple way by considering the equation where v is small. The solution is of type 7] = A -f JBe*>, where A and B are constants. When the term on the right of (IT) is omitted the solution is simply 77 — A. When x and v are both small and positive the term Bexlv, which is omitted in the foregoing method of approximation, may be really the dominant term. In this example all the terms involving derivatives of the highest order have been omitted, and as a general rule this is more dangerous than the omission of only some of the terms as in the case of the vibrating rod. The omission of the second term from the rod equation seems to be quite justifiable when the rod is very thin. When the rod is thick Timoshenko's theory* shows that there is a term giving the correction for shear which is at least as important as the second term of the usual equation (I). This point relating to the danger of omitting terms involving derivatives of the highest order comes up again in hydrodynamics when the question .of the omission of some or all of the viscous terms comes under considera- tion. The omission of all the viscous terms lowers the order of the equations and requires a modification of the boundary conditions. This does riot lead to very good results. On the other hand, in PrandtFs theory of the boundary layer some of the viscous terms are retained, the boundary condition of no slipping at the surface of a solid body is also retained and the results are found to be fairly satisfactory. EXAMPLE r» Au j. ^u A- &v Bu , du Prove that the equations - = a - - -f b ~- , ox ox oy dv du , du 2 = c 5- -f d 5- oy ox oy give an equation of the second order which is elliptic, parabolic or hyperbolic according as (a — d)2 + 46c is less than, equal to or greater than zero. [E. Picard, CompL Rend. t. cxn, p. 685 (1891).] § 1-71. Potentials and stream-functions. The classical equations are of great mathematical interest and have played an important part in the * Phil. Mag. (6), vol. XLI, p. 744 (1921). The equation used by Timoshenko is of type „ 23S? d*T) 2 EK 3~4 + P aTa - P" * r * where jz is the mocjultis of rigidity and a is a constant which depends upon the shape of the cross - section. For the equation of resisted vibrations see Note II, Appendix. 78 The Classical Equations development of mathematical analysis by suggesting fruitful lines of investigation. It can be truly said that the modern theory of functions owes its origin largely to a study of these equations. The theory of functions of a complex variable is associated, for instance, with the theory of con- jugate functions and the solutions of Laplace's equation. If, for instance, we write 0 + ty=/(a; + ty) =/(«), where/ (z) is an analytic function* and (f> and 0 are real when x and y are real, we have, for points in the domain for which/ (z) is analytic, = dy dy J where f (z) denotes the derivative of /(z). These equations give Equating the real and imaginary parts of the two sides of this equation, we see that ~ . - , d<f> dJj V= - ar = ^, say. oy ox J These relations between the derivatives of two conjugate functions <f> and ifj are called Cauchy's relations because they play a fundamental part in Cauchy's theory of functions of a complex variable. The relations can also be given many very interesting physical interpretations. The simplest from a physical standpoint is, perhaps, that in which u and v are regarded as the component velocities in the plane of x, y of a particle of a fluid in two-dimensional motion, the particle in question being the particular one which happens to be at the point (x, y) at time t. If u and v are independent of t the motion is said to be "steady" and a curve along which it is constant may be regarded as a "stream-line" or "line of flow" of the particles of fluid. The condition that a particle of the fluid should move along such a line is, in fact, expressed by the differential equations 7 , d^=d-y=dt ...... (B) u v v ' which give vdx — udy = 0, that is dift = 0 or if/ = constant. * The reader is supposed to possess some knowledge of the properties of analytic functions. Conjugate Functions 79 Another way of looking at the matter is to calculate the "flux" across any line AP from right to left. This is expressed by the integral where ds denotes an element of length of AP and the suffix is used to indicate the point at which \fj is calculated. It is clear from this equation that there is no flow across a line AP along which if/ is constant. The conjugate function <£ is called the " velocity potential" and was first introduced by Euler. The curves on which (f> is constant are called " equipotential curves." The function </f is called the stream-function or current function, it was used in a general manner by Earnshaw. It must be understood that the fluid motion which is represented by such simple formulae is of an ideal character and is only a very rough approximation to a real motion of a fluid. A study of this type of fluid motion serves, however, as a good introduction to the difficult mathe- matical analysis connected with the studies of actual fluid motions. It will be worth while, then, to make a few remarks on the peculiarities of this ideal type of fluid motion*. In the first place, it should be noticed that the expression udx 4- vdy is an exact differential dcf>, and so the integral I (udx + vdy) represents the difference between the values of <f> at the ends of the path of integration. If the function (f> is one -valued the integral round a closed curve is zero, but if <f> is many-valued the integral may not vanish. The value of the integral in such a case is called the circulation round the closed curve. It is different from zero in the case when <£ -f ty = i log z = i (log r + id) and the curve is a circle whose centre is at the origin. In this case <£ = _ 0, 0 = log r, and it is easily seen that the circulation F defined by the integral f r r2zr r = udx + vdy = ld<f> = - dO J J Jo is equal to — 2n. The fluid motion for which <£ -f ty = — A log z, where A is a constant, is said to be that due to a vortex of strength F when ir A is an imaginary quantity 0- . If, on the other hand, A is real, the motion J-tTT is said to be due to a source if — A is positive and due to a sink if — A is negative. The flow in the last two cases is radial. 80 The," Classical Equations Since the stream -function in the last two cases is — Ad and is not one- valued, the flux across a circle whose centre is at 0 is — 2-n-A. The flow due to a vortex, source or sink at a point other than the origin may be represented in the same way by simply interpreting r and 6 as polar co-ordinates relative to the point in question. Since the equations expressing u and v in terms of </> and $ are linear, the component velocities for the flow due to any number of vortices, sources and sinks may be derived from the complex potential <f> + it- 27T s («• - *&) log IX- x* + i(y- y*)] > where the constants as , j8s specify the strengths of the source and vortex associated with the point (xs, ys). The word source is used here in a general sense to include both source and sink. One further remark may be made regarding the motion if we are interested in the career of a particular particle of fluid. If x0, y0 are the initial co-ordinates of this particle at time t these quantities at time t will be functions of x, y and t %» - / (x, y, 0» 2/0 = g (x> y, 0> but functions of such a nature that the equations (B) are satisfied when xQ and y0 are regarded as constant. We have then «!/+«!/+ jjf-o, «*+, <*+*=, o ...... (c.D) dx oy dt ox oy ot ' and any quantity h which can be expressed in the form h = F (XQ , y0) will be a solution of the equation dh dh dh a. + M 3 + V 5 - = 0, dt dx dy and will be constant throughout the motion. We shall write this equation in the form dhjdt = 0 and shall call dh/dt the complete time derivative of h. When the motion is steady we evidently have difj/dt = 0. The equations (C) and (D) may be solved for u and v if —-- = 1 and o (Xj yj give expressions *~ which satisfy the equation = ox dy on account of « - > = 1. a (x, y) a (x, y) This last equation expresses that the area occupied by a group of particles remains constant during the motion. To obtain a solution of this equation we take x and XQ as new independent variables, then dy ^ d (x, y) = 8 (x, y) 3 (x^y^) ^ d_(x^ ,J/Q) dxQ d(x, x0) 3 (x, *0) 9 (x, y) 9 (x, x0) ' Motion of a Fluid 81 , dy dyQ and so ^- = — /°. OXQ dx This means that ydx — yQdxQ is an exact differential and so we may write where F = F (x, x0, t) and t is regarded as constant. If, however, we allow t to vary and use brackets to denote derivatives when x} y and t are regarded as independent variables, we have dyQ d*F S2F A — "U — />/ i V 1± ^ 'S 'S I" <•*» " <•> . J «£ ratoft OXnCt O-i* i- , -^fog^ \Jfo-; > 1 = 3y/ " 3a:8x0 _ _ fov cte^y 9#9£ "^ 9#0 /3a?0\ = — u. t\cy / O 17f Hence we may write 0 = — — and obtain a convenient expression for the stream-function. Another physical interpretation of the functions <f> and $ is obtained by regarding <f> as the electric potential and u, v as the components of the electric field strength due to a set of fictitious point charges, or, if we prefer a three-dimensional interpretation, to a system of uniform line charges on lines perpendicular to the plane of x, y. The curves cf> = constant are then sections by this plane of the equipotential surfaces cf> = constant, while the curves 0 = constant are the "lines of force" in the plane of x, y. For brevity we shall sometimes think in terms of the fictitious point charges and call a curve <f> = constant an "equipotential." Again, <f> may be interpreted as a magnetic potential of a system of magnetic line charges (fictitious magnetic point charges) or of electric currents of uniform intensity flowing along wires of infinite length at right angles to the plane of x, y. The curves $ = constant are again lines of force, a line of force being defined by the equations dx _dy u v 82 The Classical Equations In all cases the lines of force are the orthogonal trajectories of the equi- potentials, as may be seen immediately from the relation dx dx dy dy ~~ ' which is a consequence of Cauchy's relations. For any number of electric or magnetic line charges perpendicular to the plane of x, y we have by definition <f> + iif* = 22^s log [x - xs + i (y - y,)], where /x8 is the density per unit length of the electricity, or magnetism as the case may be, on the line which passes through the point (x, y)\ It must be understood, of course, that when </> is the electric potential we consider only electric charges and when </> is the magnetic potential we consider only magnetic charges. When the number of terms in the series is finite we can certainly write . . . , . , J </» + *0=/(x + ty) =/(«), where / (z) is a function which is analytic except at the points z = zs . When in the foregoing equation p,8 is regarded as a purely imaginary quantity, <f> may be interpreted as the magnetic potential of a system of electric currents flowing along wires perpendicular to the plane of x, y. If fjis = iC8 the current along the wire xs , ys is of strength C8 and flows in the positive direction, i.e. the direction associated with the axes Ox, Oy by the right-handed screw rule. When a potential function <f> is known it is sometimes of interest to determine the curves along which the associated force (or velocity) has either a constant magnitude or direction. This may be done as follows. We have log (u - iv) = log/' (x + iy) = <X> + f¥, say, where O = £ log (u2 + v2), T = TT — tan*1 (v/u). The curves O == constant are clearly curves along which the magnitude (u2 + V2)* of the force or velocity is constant, while Y = constant is the equation of a curve along which the direction of the force is constant. The functions <t> and T are clearly solutions oi Laplace's equations, i.e. 320 32O _ dx2 + dy2 ~ A function O which satisfies this equation is called a logarithmic potential to distinguish it from the ordinary Newtonian potential which occurs in the theory of attractions. The electric and magnetic potentials of line charges are thus logarithmic potentials. Two-Dimensional Stresses 83 A logarithmic potential <D is said to be regular in a domain D if ao ao ' > ' 2' are continuous functions of x and y for all points of D. If D is a region which extends to infinity it is further stipulated that lim <f> (x, y) = (7, lim r - ^ = lim r ~ - = 0, r — > oo r — >• oo 0«£ / — > oo 0y (r2 - a:2 + y*), where C is a finite quantity which may be zero. In this sense the potential of a single line charge is not regular at infinity. Still another physical interpretation of conjugate functions is obtained by writing Y y JL y y ,/, Ax = — I y = 0, Ay = I X = </f. Cauchy's relations then give These are the equations for the equilibrium of an elastic solid when there are no body forces and the stress is two-dimensional. The quantities (Xx, Xv) are interpreted as the component stresses across a plane through (x, y) perpendicular to the axis of x, while (Yx, Yv) are the component stresses across a plane perpendicular to the axis of y. The relation Xv = Yx is quite usual but the relation Xx + Yy — 0 indicates that the distribution of stress is of a special character. A stress system satisfying this condition can, however, be obtained by writing *„-- rv=£(«2-»2), Yx = Xv=*-uv, for these equations give SXX dXv /du dv\ /dv du The fact that the various potentials <f> and 0 which have been considered so far are solutions of Laplace's equation is a consequence of the circumstance that they have been defined as sums of quantities that are individually solutions of this equation. No physical principle has been used except a principle of superposition which states that when the individual terms give quantities with a physical meaning, 6-2 84 The Classical Equations the sum will give a quantity with a similar physical meaning. In the analysis of many physical problems such a superposition of individual effects is not strictly applicable, for the sources of a disturbance cannot be supposed to act independently, each source may, in fact, be modified by the presence of the others or may modify the mode of propagation of the disturbance produced by another. Such interactions will be left out of consideration at present, for our aim is not to formulate at the outset a complete theory of physical phenomena but to gradually make the student familiar with the mathematical processes which have been used successfully in the gradual discovery of the laws of physical phenomena. In applied mathematics the student has always found the formulation of the fundamental equations of a problem to be a matter of some difficulty. Some men have been very successful in formulating simple equations be- cause, by a kind of physical instinct, they have known what to neglect. The history of mathematical physics shows that in many cases this so-called physical instinct is not a safe guide, for terms which have been neglected may sometimes determine the mathematical behaviour of the true solution. In recent years the tendency has been to try to work with partial differential equations and their solutions without the feeling of orthodoxy which is created by a derivation of the equations that is regarded for the time being as fully satisfactory. The mathematician now feels that it is only by a comparison of the inferences from his equations with the results of ex- periment and the inferences from slightly modified equations that he can ascertain whether his equations are satisfactory or not. In the present state of physics the formulation of equations has not the air of finality that it had a few' years ago. This does not mean, however, that the art of formulating equations should be neglected, it means rather that mathematicians should also include amongst their special topics of study the processes which lead to the most interesting partial differential equations of physics. These pro- cesses are of various kinds. Besides the process of elimination from equa- tions of the first order there are the methods of the Calculus of Variations and methods which depend upon the use of line, surface and volume integrals. Mathematically, the direct process of elimination is the simplest and will be given further consideration in § 1-82. / § 1-72. Geometrical properties of equipotentials and lines of force. When the potential </> is a single-valued function of x and y there cannot be more than one equipotential curve through a given point P in the (x, y) plane. An equipotential curve <£ = </>0 may, however, cross itself at a point and have a multiple point of any order at a point PQ. In such a case the tangents at the multiple point are arranged like the radii from the centre to the corners of a regular polygon. To see this, let us take the origin at P0, Method of Inversion 85 then the terms of lowest degree in the Taylor expansion of </> — <£0 are of yP cnenla- (x 4- iy)n 4- cne~nt(l (x — iy)n, where n is an integer and c and a are constants. In polar co-ordinates x = r cos 9, y = r sin #, these terms become 2cnrn cos ^ (0 4- «), and the directions of the n tangents are given by cos n (6 -f «). The possible values of n (9 4- a) are thus ?r/2, 37r/2, ... (n — J) TT, the angle between con- secutive tangents being TT/U. Since cos n (9 4- a) is positive for some values of 6 and negative for others, the function <f> cannot have a maximum or minimum value at a point, for this point may be chosen for origin and the expansion shows that there are points in the immediate neighbourhood of the origin for which </> > (f)Q , and also points for which <£ < </>0 . By means of the transformation x' - iy' = k* (x 4- iy)-1, x' -f ^y = &2 (x - iy)~l, which represents an inversion with respect to a circle of radius k and centre at the origin, an equipotential curve of a system of line charges is transformed 1S' into an equipotential curve of another system of line charges. In polar co-ordinates we have r' = *Yr> 9' = °- If in Fig. 10 Q corresponds to P and B to A, we have Rjr = B'/b, where AP = R, OP = r, BQ - R', OB = b. For a number of points A and the corresponding points B S/t. log (R,/r) = SMs (log «' - log b). An equipotential system of curves represented by the equation is thus transformed into an equipotential system represented by the equation = C _ ^^ log 6. A line charge at B is seen to correspond to a line charge of equal strength at A and another one of opposite sign at 0 which may be supposed to correspond to a line charge at infinity sufficient to compensate the charge at B. Ap equipotential curve with a multiple point at O inverts into an equi- potential which goes to infinity in the directions of the tangents at the 86 The Classical Equations multiple point. This indicates that the directions in which an equipotential goes to infinity are parallel to the radii from the centre to the corners of a regular polygon. In the simple case of two equal line charges at the points (c, 0), (— c, 0) the equipotentials are log Rl + log R2 = constant, or R^2 = a2, where a is constant for each equipotential. These curves are Cassinian ovals with the polar equation r4 -f c4 - 2r2c2 cos 20 = a4. When a = c we obtain the lemniscate r2 = c2 cos 26 with a double point at the origin. The tangents at the double point are perpendicular. Inverting we get the equipotentials for two equal line charges of strength + 1 at the points (6, 0), (- 6, 0), where be = k2 and a line charge of strength — 2 at the origin. The equipotentials are now log RI + log jR2' - 2 log r' = constant, or c*R/jR2' = aV2. Dropping the primes we have the polar equation C2 (r4 + C4 _ 2r2c2 cos 20) - a*r4 of a system of bicircular quartic curves. When a = c we obtain the rect- angular hyperbola r2 cos 20 = c2 which is the inverse of the lemniscate. The rectangular hyperbola goes to infinity in two perpendicular directions. It is easily seen that lines of force invert into lines of force. In Fig. 10, if we denote the angles POA, PAB, QBO by d, Q and 0' respectively, we have the relation _ Hence the lines of force represented by the equation S fj,s ©/ = constant transform into the lines of force represented by the equation 2/A,05 — 02/is = constant. In particular, the lines of force of two equal line charges ©i H- ©2 = constant, being rectangular hyperbolas, invert into the family of lemniscates repre- sented by _ and these are the lines of force of two equal line charges of strength -f 1 and a single line charge of strength - 2 at 0. At a point of equilibrium in a gravitational, electrostatic or magnetic field, the first derivatives of the potential vanish and so the equipotential curve through the point has a double point or multiple point. A similar Lines of Force 87 remark applies to a curve ifj = constant, but this curve cannot strictly be regarded as a single line of force for, if we consider any branch which passes through the point of equilibrium without change of direction, the force is in different directions on the two sides of the point of equilibrium and the neighbouring linqs of force avoid the point of equilibrium by turning through large angles in a short distance. This is exemplified in the case of two equal masses or charges when the equipotentials are Cassinian ovals which include a lemniscate with a double point at the point of equilibrium. The lines of force are then rectangular hyperbolas, the system including one pair of perpendicular lines which cross at the point of equilibrium. In plotting equipotential curves and lines of force for a given system of line charges it is very useful to know the position of the points of equi- librium, since the properties just mentioned can be employed to indicate the behaviour of the lines of force. At a point of stagnation in an irrota- tional two-dimensional flow of an inviscid fluid the component velocities vanish and so the first derivatives of the velocity potential and stream- function are zero. The properties of the equipotentials and stream-lines at a point of stagnation are, then, similar to those of equipotential and lines of force at a point of equilibrium. There is, however, one important difference Between the two cases. In the electric problem the field is often bounded by a conductor, i.e. an equipotential surface, while in the hydro- dynamical problem the field of flow is generally bounded by some solid body whose profile in the plane z = 0 is a stream-line. A point of stagnation frequently lies on the boundary of the body and two coincident stream- lines may be supposed to meet and divide there, running round the body in opposite directions and reuniting at the back of the body when the profile is a simple closed curve. A point on a conductor may be a point of equilibrium if the conductor's profile is a curve with a double point with perpendicular tangents or if it consists of two curves cutting one another orthogonally at all their common points. It should be remarked, however, that the force at a double point may be either zero or infinite ; it is zero when the double point repre- sents a pit or dent in the curve, but is infinite when the double point represents a peak. This may be exemplified by the equations <f> = x2 — y2, iff = 2xy. If the field lies in the region x > 0, y2 < x2, the force is zero at 0 and there is a single line of force through 0, namely, y = 0 (Fig. 11). If, on the other hand, the field is outside the region x < 0, x2 > y2, and , _ x2-y2 _ 2xy the force at the origin is infinite for most methods of approach and there are three lines of force through the origin (Fig. 12). Similarly, when two conductors meet at any angle less than ?r, but a submultiple of TT, the angle being measured outside the conductor. The 88 The Classical Equations point of intersection is a point of equilibrium and we have the approximate <£ = 2cnrn cos n (6 -f «) expression of Force Line of Force Fig. 12. for the value of </> in the neighbourhood of the point, the equation of the conductor in the neighbourhood of the point being n (9 -f «) = ± n/2 and the field being in the region — 9 < n (9 -f a) < ~ . The angle is in this case £ £ 7T/n and the radial force ^ varies initially according to the (n — l)th power of the distance as a point recedes from the position of equilibrium. The corresponding approximate expression for «/r is ifj = 2cnrn sin n (6 + a) and there is a single line of force 9 = — a which lies within the field, this being its equation in the immediate neighbourhood of the point O. There is another simple transformation which is sometimes useful for deriving the equipotentials and lines of force of one set of line charges from those of another. This is the transformation z' = z + a2/z which gives two values of z for each value of z' . Let these be z and z, then zz — a* Similarly, let z1 and zl correspond to z/, then z' - z/ - z - zl -f a*/z - a2/*! = (z - zj (1 - «2 = (Z - 2J (1 - V) = (Z - Z,) (Z - Zj/Z. Taking the moduli we obtain a relation where r = z - z r = Geometrical Properties 89 Similarly, if z2 and z2 correspond to z2' we have, with a similar notation, < = r2r2fr, j r2x r2r2 and so -~ = -~~ . ri Wi The transformation thus enables us to derive the equipotentials for four charges (1, 1, — 1, — 1) from the equipotentials for two charges (1, — 1) and a similar remark holds for the lines of force, as may be seen by equating the arguments on the two sides of the equation This is just one illustration of the advantages of a transformation. A general theory of such transformations will be developed in Chapter III. Some geometrical properties of equipotential curves and lines of force may be obtained by using the idea of imaginary points. The pair of points with co-ordinates (a ± ij8, b T ia) are said to be the anti-points of the pair with co-ordinates (a ± a, b ± /?), the upper or lower sign being taken throughout. Denoting the two pairs by Fl , F2 ; Sl , S2 respectively, we can say that if S1 and S2 are the real foci of an ellipse, then Ft and F2 are the imaginary foci. Fl and F2 can also be regarded as the imaginary points of intersection of the coaxial system of circles having 8l and S2 as limiting points. If the co-ordinates of F± and F2 are (xl9 yj, (x2, y2) respectively arid those of Slt S2 are (£19 T?^, (£2, 7?2) respectively, we have xi + iyi = a + * 4- a -f ij8 = & + iifr, x\ - iy\ = a ~ ib - a + ip = ^2 - i^2, xz + iy* = a + ib - « - *j3 = ^2 -f i^2, ^2 ~ *2/2 = a — i& + a — *'j8 = f ! — i-^! . If now i£ -f- iv — f (x + iy), u — iv — f (x — iy), and $! , $2 lie on a curve u = constant, we have The foregoing relations now show that / te + iyi) + f fa - iy*) = / (*2 + and this means that Fl9 F2 lie on a curve v = constant. When the imaginary points on a curve v = constant admit of a simple geometrical representation or description, the foregoing result may be sometimes used to find the curves u — constant. If the curve v = constant is a hyperbola, the imaginary points in which a family of parallel lines meet the curve have geometrical properties which are sufficiently well known to enable us to find the anti-points of each pair of points of intersection. 90 The Classical Equations These anti-points lie on a confocal ellipse which is a curve of the family u = constant. By taking lines in different directions the different ellipses of the family u — constant are obtained. Similarly, by taking a set of parallel chords of an ellipse and the anti-points of the two points of inter- section of each chord, it turns out that these anti-points all lie on a confocal hyperbola, and by taking families of lines with different directions the different hyperbolas of the confocal family may be obtained. In this case the relations are particularly simple. In the general case when one curve of the family u = constant is given there will be, pre- sumably, a family of lines whose imaginary intersections with this curve are pairs of points with anti-points lying on one curve of the family v = con- stant, but these lines cannot be expected, in general, to be parallel, and a simple description of the family is wanting. EXAMPLES 1. If a family of circles gives a set of equipotential curves, the circles are either con- centric or coaxial. 2. Equipotentials which form a family of parallel curves must be either straight lines or circles. [Proofs of these propositions will be found in a paper by P. Franklin, Journ. of Math. and Phys. Mass. Inst. of Tech. vol. vi, p. 191 (1927).] § 1-81. The classical partial differential equations for Euclidean space. Passing now to the consideration of some partial differential equations in which the number of independent variables is greater than two we note here that the most important equations are Laplace's equation 32F 32F 32F „ the wave-equation ^v 92F 92F 1 92F "S*» + ay« + a*'2 ^c2 w the equation of the conduction of heat the equation for the conduction of electricity dE ^ ....... (D) and the wave-equation of Schrodinger's theory of wave -mechanics. This last equation takes many different forms and we shall mention here only the simple form of the equation in which the dependence of 0 on the time has already been taken into consideration. The reduced equation is then . 8?+^( ^= ' ...... ( } where F is a function of x, y and z and E is a constant to be determined. Laplace's Equation 91 In these equations K represents the diffusivity or thermometric con- ductivity of the medium, K the specific inductive capacity, /x the per- 'meability, and a the electric conductivity of the medium. The quantities c and h are universal constants, c being the velocity of light in vacuum and h being Planck's constant which occurs in his theory of radiation. Laplace's equation, which for brevity may be written in the form V2V = 0, may be obtained in various ways from a set of linear equations of the first order. One set, dv dv dv sx dY dz x=s*' Y=dy> z = ~dz' a*+ay + al = 0' (F) occurs naturally in the theory of attractions, V being the gravitational potential and X, Y, Z the components of force per unit mass. The last equation is then a consequence of Gauss's theorem that the surface integral of the normal force is zero for any closed surface not containing any attracting matter. The same equations occur also in hydrodynamics, the potential V being replaced by the velocity potential <£ and the quantities X, Y, Z by the component velocities u, v, w. The equation is then the equation of con- tinuity of an incompressible fluid. The electric and magnetic interpretations of X , Y, Z and V are similar to the gravitational except that the electric (or magnetic) potential is usually taken to be — V when X , Y , Z are the force intensities. As in the two-dimensional theory, Laplace's equation is satisfied by the potential V because by the principle of superposition V is expressed as the sum of a number of elementary potentials each of which happens to be a solution of Laplace's equation, the elementary potential being of type V=[(x- x')* +(y- y')2 + (a - z')T} = l/S. When V is interpreted as the electrostatic potential this elementary potential is regarded as that of a unit point charge at the point (#', y* ', z') ; when V is interpreted as a magnetic potential the elementary potential is that of a unit magnetic pole. In the theory of gravitation the elementary potential is that of unit mass concentrated at the point (#, yy z). A more general expression for a potential is V = 2m5 [(x - *.)« + (y - ys)* + (z - ».)•]-*, where the coefficient ms is a measure of the strength of the charge, pole or mass concentrated at the point (x8, y8, zs). If we write <f> in place of V, where <f> is a velocity potential for a fluid motion in three dimensions, the elementary potential is that of a source and the coefficient ms can be interpreted as the strength of the source at (x3, y8, zs). Sources and sinks 92 The Classical Equations are useful in hydrodynamics as they give a convenient representation of the disturbance produced by a body when it is placed in a steady stream. § 1-82. Systems of partial differential equations of the first order which lead to the classical equations. When we introduce algebraic symbols x ~ dx ' v ~~ dy ' z ~~ dz for the differential operators the equations (F) of § 1-81 become X - Dx V = 0, 7 - Dy V = 0, z - A v = o, and the algebraic eliminant 1 0 0 -A 0 1 0 -A 0 0 1 -A D A X) 0 - 0 is simply Z>x2 -f Z>v2 4- A2 = 0. If, on the other hand, we consider the set of equations dw dv ds _ dy dz dx ' du dw ds — - , _ r= {J dz ox dy dv du ds _ 3# 3?/ 82 ~ ' du dv dw which give V2u = V2v = V2w; = V2*s = 0, the corresponding algebraic equations .(G) give the eliminant - AM + Af Dvw- Dxs = 0, A* = °> A« = o, AM + A« + Dzw = 0 0 -A Dy -A = o, A 0 -A -A -A -DX 0 -A A />« A 0 which is equivalent to + + •(H) (I) Elastic Equilibrium 93 These examples show that the problem of finding a set of linear equa- tions of the first order which will lead to a given partial differential equation of higher order admits a variety of solutions which may be classified by noting the power of the complete differential operator (in this case (V2)2) which is represented by the algebraic eliminant written in the form of a determinant. It is known that Laplace's equation also occurs in the theory of elas- ticity. If u, v, w denote the components of the displacements and Xx, Yy, Zz, Yz, Zx, Xy the component stresses the equations for the case of no body forces are .(J) * 1 a# v ay + u^\. z Tz'~ o, dYx dYy , dYz A dx dy ' dz U) ™*4 dZv SZZ_ 0, and if the substance is isotropic the relations between stress and strain take the form ~ = AA .(K) where V r/ [ ^™ i 7--z' = *(% + &)• du dw* + dx du . _ du dv dw dx dy dz ' The equations obtained by eliminating Xx, Yy, Zz, Yz, Zx, Xv are = (A + ^)~, = (A + 11.) ' and, except in the case when A + 2/* = 0, a case which is excluded because A and fj. are positive constants when the substance is homogeneous, these 94 The Classical Equations equations imply that A is a solution of Laplace's equation. The algebraic eliminant is in this case (D,2 + A,2 + A2)3 = 0. ...... (L) It is easily seen that the quantities XX9 Yy, Zz, 7C, Zx, Xy, u, v, w are all solutions of the equation of the fourth order V2V2w = 0, i.e. V4^ = 0, which may be called the elastic equation. The algebraic equation obtained by eliminating the twelve quantities Xx, Xy, Xz, YX) Yy, Yz, Zx, Zy, Zz, u, v, w from the twelve equations (J) and (K) by means of a determinant is also equivalent to (L). The question naturally arises whether as many as four equations are necessary for 'the derivation of Laplace's equation from a set of equations of the first order. The answer seems to be yes or no according as we do or do not require all the quantities occurring in the linear^quations of the first order to be real. Thus, if we write U = u — iv, V = w + is, where u, v, w, s are the quantities satisfying the equations (H), it is easily seen that du .du dv A dv .dv du „ . 1 4 . ___ I __ - I I _ _ n _ A ~liT~ I V *\ I ?N - VJ ~f\ V f* ~ - ~^. ~ - U. dx oy dz ex cy dz and these equations imply that = 0, V2F= 0. The algebraic eliminant is in this case simply (G). It should be noticed that if we write ict in place of y the two-dimensional wave-equation may be derived from the two equations dU dV 13^_0 ?Z_^_l?Z_n dx + dz + c ~dt ~ ' dx dz c~dt~ which have real coefficients. The wave-equation (B) may also be derived from two linear equations of the first order dU .dU_dV IdV dx + l dy ~ ~fa + c dty w__idv^i su_du dx dy "~ c dt dz ' but in this case the coefficients are not all real. The algebraic eliminant is in this case simply * c2 (IV + Dv2 -f A2) - A2 = 0. Equations leading to the Wave-Equation 95 To obtain the wave-equation from a set of linear equations of the first order with only real coefficients we may use the set of eight linear equations, dy dz~ 'ds dx' dy dz ~dx ds' da dy_ dY dT dx dz _dr dp dz dx 'ds dy' 'dz dx dy ds' dp da_ dZ dT dY dx dr dy dx~ dy~ ds dz' 'dx dy ~dz ds' X i 57_ dT dz da dp dr dy Jx + ' d~y~ ds dz' dx4 'dy ~ds ~dz' in which for convenience s has been written in place of ct. These equations imply that X , Y, Z, T, a, j3, y and r are all solutions of the wave-equation. The algebraic eliminant is now 0* = (Dx* + Dv* + A2 - A2)4 = 0. If in the foregoing equations we put T = r = 0 we obtain a set of equations very similar to that which occurs in Maxwell's electromagnetic theory. The eight equations may be divided into two sets of four and an algebraic eliminant. may be obtained by taking three equations from each set and eliminating the six quantities X, 7, Z, «, /?, y. There are altogether sixteen possible eliminants but they are all of type 02L = 0, where the last factor L is obtained by multiplying a term from the first of the two D. Dv A D. Dx Dy Dz D8 by a term from the second. §1-91. Primary solutions. Let/ (&, £2, ... £w) be a homogeneous poly- nomial of the degree in its m arguments & , £2 > • • • £w an<^ ^e^ each of the quantities D8 that is used to denote an operator d/dx8 be treated as an algebraic quantity when successive operations are performed. The equation /(A,A,-A»)*=o (A) is then a linear homogeneous partial differential equation of a type which frequently occurs in physics. An equation such as Dfw = D2w may be included among equations of the foregoing type by writing u = e** . w, and noting that u satisfies the equation (A2 ~ A A) u = 0. A solution of the form 96 The Classical Equations in which 0X, 02, ... 0, are particular functions of.a^, x2, x3, ... xm, and F is an arbitrary function of the parameters 019 02, 03, ... 0>s., will be called a primary solution. An arbitrary function will be understood here to be a function which possesses an appropriate number of derivatives which are all continuous in some region R. Such a function will be said to be con- tinuous (/), n) when derivatives up to order n are specified as continuous. It can be shown that the general equation (A) always possesses primary solutions of type u = F (0), ...... (B) where 6 == c,^ 4- c2x2 -f ... cmxm, ...... (C) and ct, c2, ... cm are constants satisfying the relation /(q,^, ...CTO)-= 0. ...... (D) This relation may be satisfied in a variety of ways and when a para- metric representation ~ , c = cro ~ C'm (ai> «2> ••• am-2 is known for the cD-ordinates of points on the variety whose equation is represented by (D), the formulae (B) and (C) will give a family of primary solutions. When ra = 2 there is generally no family of primary solutions but simply a number of types, thus in the case of the equation (A2 - A2) u = o there are the two types u = F (^ -f a;2), te = J7 (^ - x2). Primary solutions may be generalised by summing or integrating with respect to a parameter after multiplication by an arbitrary function of the parameter. Thus in the case of Laplace's equation we have a family of primary solutions V = F (0) . G (a), where 0 = z -f ix cos a -f iy sin «, and a is an arbitrary parameter. Generalisation by the above method leads to a solution which may be further generalised by summation over a number of arbitrary functional forms for F (0) and G (a) and we obtain Whittaker's solution * f2ir V = W (z -f ix cos a -f iy sin a, a) da, .'o which may also be obtained directly by making the arbitrary function F a function of a as well as of 0. The primary solutions (B) are not the only primary solutions of * Math. Ann. vol. LVII, p. 333 (1903); Whittaker and Watson, Modern Analysis, ch. xvin. Primary Solutions 97 Laplace's equation, for it was shown by Jacobi* that if 9 is defined by the equation 0 - # (9) + *, (8) + * (9), ...... (F) where £ (6), rj (0) and £ (0) are functions connected by the relation [f W]2 + h (0)]2 + K Wl2 = o, then F = .F (8) is a solution of Laplace's equation. This is easily verified because iff M=l-x? (d) - yr,' (d) - z? (6), we have These equations give M ~ ' [*" (•> + OT" w + *r (»n - f <»> - and so V20 - 0, V2 {F (0)} - 0. This theorem is easily generalised. If ca (a), c2 (a), ... cw (a) are functions connected by the identical relation (D) the quantity 6 defined by the equation & = ^ ^ + ^^ (&) + _ ^^ ((?) (Q) is such that i/ = ^ (0) is a primary solution of equation (A). Since v = du/dxl is also a solution of the same differential equation it follows that if G (6) is an arbitrary function and M = 1 - *lC/ (6) - x2c2' (6) - ... xmcm' (0) tne expression v = M~1G (0) is a second solution of the differential equation. The reader who is familiar with the principles of contour integration will observe that this solution may be expressed as a contour integral If O (a) da V — — i L 2-rri ) c a - X& (a) - X2c2 (a) - ... xmcm (a) ' where C is a closed contour enclosing that particular root of equation (G) which is used as the argument of the function G (0). It is easy to verify that the contour integral is a solution of the differential equation because the integrand is a primary solution for all values of the parameter a and has been generalised by the method already suggested. * Journal fiir Math. vol. xxxvi, p. 113 (1848); Werke, vol. IT, p. 208. f We use primes to denote differentiations with respect to 0. 98 The Classical Equations In this method of generalisation by integration with respect to a para- meter the limits of integration are generally taken to be constants or the path of integration is taken to be a closed contour in the complex plane. It is possible, however, to still obtain a solution of the differential equation when the limits of integration are functions of the independent variables of type 0. Thus the integral re V = W (z + ix cos a + iy sin a, a) da J o V2F = 0 when 6 is < (a) _ 77 (a) _ £ (a) satisfies Laplace's equation V2F = 0 when 0 is defined by an equation of type (F) where icosa ism a 1 When the equation (A) possesses primary solutions of type u = F (6) and no primary solutions of type u = F (0, <f>) it will be said to be of the first grade. When it possesses primary solutions of type u = F (0, <f>) and no primary solutions of type u = F (0, <f>, if/) it will be said to be of the second grade and so on. The equation d2u/dxdy = 0 is evidently of the first grade because the general solution is u = F (x) -f G (y), where F and G are arbitrary functions. The primary solutions are in this case F (x) and G (y). Laplace's equation V2 (u) = 0 is also of the first grade but the equation du Su du _ dx+dy+dz** is of the second grade because the general solution is of type u = F (y - z, z - x). There is, of course, a primary solution of type u= F (y- z,z- x,x- y), where F (0, </>, 0) is an arbitrary function of the three arguments 0, <f>y ^r, but these arguments are not linearly independent ; indeed, since 0 + <£ -f ^ = 0, a function of 0, <f> and ifi, is also a function of 0 and <f>. In the foregoing definition of the grade of the equation it must be understood, then, that the parameters 0, <f>, $, etc., are supposed to be functionally independent. The differential equation has not usually a grade higher than one. If, in particular, an attempt is made to find a solution of type where 0 = x^ + x2& -f z3£3 -f a?4f4, <f> = #!*?! + #2*72 + *3>?3 + Primary Solution of the Wave-Equation 99 it is found that a number of equations must be satisfied. These equations imply that where a and b are arbitrary parameters and this means that all points of the line lie on the surface whose equation is /(r r Y v \ — 0 V^i> ^2) ^3? •*'4; ~~ u« When / (#! , x2 , x3 , x4) has a linear factor of the first degree or is itself of the first degree the equation (A) is of grade 3. In particular the equation possesses the general solution Tfl / \ and so is of grade 3. An equation with m independent variables which, by a simple change of variables, can be written in the form a / a a a is said to be reducible. Such an equation is evidently of grade m — 1. It is likely that whenever the number of independent variables is m and the grade m — 1 the equation is reducible. The wave-equation is of grade 2 because there is a primary solution of type u = F(0, <£), where n .... 0 — x cos a 4- i/ sin a -f tz, q> = x sin a. — y cos a 4- This solution may be generalised so as to give a solution u= ! 'F (0,<f>, a) da, Jo analogous to Whittaker's solution of Laplace's equation. Theequations ,,_ c St ~ ' _. ~dx S^c St 7-2 100 The Classical Equations which may be written in the abbreviated form * and which give the simple equations of Maxwell curl 17 =- - 3j^, div# = 0, C vt when the vector Q is replaced by H + iE, where E and // are real, may be satisfied by writing where q (a) is a vector with components cos «, sin a, i, respectively and F (0, <f>, a) is an arbitrary function of its arguments. EXAMPLES 1. Let £, v), £, T he functions of a, /9, y connected by the relation e + ^ + j2 + r2 = i, and let X =-- rx - fy + rjz — ft + u, Z = — yx + (y + TZ — & + w, Y = & + ry- fz-iit + v, T = (x + ijy + ^2 f- T< + «. Prove that if the integration extends over a suitable fixed region the definite integral V satisfies the differential equation 2. If V = F(A,B,C,D,E) is a solution of the equation a2F a2F a2F a2F_a^F a^2 + dB* + BC* + en2 ~ BE* when considered as a function of A, B, C, D and E; then, when A =» 2 (xs -f yw — zv — tu), G « 2 (25 + xv — yu — tw), B = 2 (2/5 4- zu — xw — tv), D = 2 (^ -f xu + yv -f zw), # = a;2 + y2 4- z2 -f ^2 -f w2 + t>2 + ^2 + *2, the function F is a solution of a27 a2F a2F a2F = a2F a2F &y a2F a^2 + at/2 + dz2 + a*2 a^2 4" a^2 + a^2 + a? when considered as a function of x, y, z, t, u, v, w, s. * We use the symbol Q to denote the vector with components Qx, Qy, Qt respectively. This abbreviated form is due to H. Weber and L. Silberstein. Characteristics 101 §"'1-92. The partial differential equation of the characteristics. It is easily seen that when 0 = c^ -f C2x2 + ... cmxm and cl9 c2, ... cm are constants satisfying the equation / (cl9 c2, ... cm) = 0, the function F (0) = u is not only a primary solution of the equation / (A , Z>2, ... Z>m) w = 0 but it is also a solution of the equation /(Aw, A*, - AH") = O. (A) This partial differential equation of the first order is usually called the partial differential equation of the characteristics of the equation /(A,A>..-An)* = o. (B) In particular, the quantity u = 0 is a solution of this differential equation and the locus 0 (xl9 x2, ... xm) = constant is a characteristic or characteristic locus of the partial differential equation. A characteristic locus can generally be distinguished from other loci of type (f> (xl9x29 ... xm) = constant by the property that it is a locus of "singularities" or "discontinuities" of some solution of the differential equation. If we adopt this definition of a characteristic locus 0 = constant it is clear that 0 = constant is a characteristic locus whenever there is a solution of the equation which involves in some explicit manner an arbitrary function F(0)9 for the function F(0) can be given a form which will make the solution discontinuous on the characteristic locus. Thus the quantity u = e~lle is a solution of the differential equation (B) when 0 =£ 0 and is discontinuous at each point of the characteristic locus 0 — 0. It should be observed that this function and all its derivatives on the side 0 > 0 of the locus 0—0 are zero for 0=0. The function u = e~llb* possesses a similar property and the additional one that the derivatives on the side 0 < 0 of the locus 0=0 are also zero. From these remarks it is evident that if there is a solution of the partial differential equation (B) which satisfies the condition that u and its derivatives up to order n — 1 have assigned values on the locus <f> (xly x2, ... xm) = constant and so gives the solution of the problem of Cauchy for the equation, this solution is not unique when <f> = 0 because a second solution may be obtained by adding to the former one a solution such as e~1/62 which vanishes and has zero derivatives at all points of the locus. This property of a lack of uniqueness of the solution of the Cauchy problem for the locus 0 (xl9 x29 ... xm) = 0 is the one which is usually used to define the characteristic loci of a partial differential equation and can be used in the case when the equation does not possess primary solutions. Since, however, we are dealing at present with equations having primary solutions the simpler definition of 0 as the argument of a primary solution or other arbitrary function occurring in a solution will serve the purpose quite well. An equation with a solution involving an arbitrary function explicitly (not under the sign of integration) will be called a basic equation. 102 The Classical Equations Let us now write p1 = D^u, p2 = D2u, ... so that the partial differential equation for the characteristics may be written in the form f(Pi>P2> ...Pm) = 0- The curves defined by the differential equations dxl dx2 dxm r lr==¥=='"T ...... dpl dp2 dpm are called the bicharacteristics * of the equation ; they are the character- istics of the equation (A) according to the theory of partial differential equations of the first order. When pl , p2 , . . . are eliminated from these equations it is found that F (dxl9dx2, ...dxm) = 0, where F (xl , x2 , . . . xm) = 0 is the equation reciprocal to/ (pl , p2 , ... pm) = 0 in the sense of the theory of reciprocal polars. In mathematical physics the loci of type u = constant, where u satisfies the equation (A), frequently admit of an interesting interpretation as wave-surfaces. The curves given by the equations (C) associated with the function u are interpreted as the rays associated with the system of wave- surfaces. In the particular case when the partial differential equation of the characteristics is d6 36 36 36 36 #-Si + u*+vt9 + W& and u, v and w are constants representing the velocity of a medium and V is another constant representing the velocity of propagation of waves in the medium, the differential equations of the bicharacteristics are dx __ dy __ dz __ dt ~H5 TTa0=='"1d0 ITs^"""^ ^Tso^Je u j* - v V v -j* - v V w ^ - v 5- -J, dt ox dt dy dt 4 oz dt and the equation obtained by eliminating x- - , ^- , ^- , -^ is (dx - udt)* + (dy - vdt)2 + (dz - wdt)* = This result is of considerable interest in the theory of sound and may be extended so as to be applicable to the case in which u, v, w and V are functions of x, y, z and t. It may be remarked that if we have a solution of (D) in the form of a complete integral . , m r & 0 = t - T - g (x, y, z, a, j3), * See J. Hadamard's Propagation des Ondes. The theory is illustrated by the analysis of § 19. Bicharacteristics and Rays 103 in which r, a and j3 are arbitrary constants, the rays may be obtained by combining the foregoing equation with the equations 39 _ A ^ - o a«-U' 3)8 ~"U' The characteristics of a set of linear equations of the first order may be defined to be the characteristics of the partial differential equation obtained by eliminating all the dependent variables except one. The relation of the primary solutions of this equation to the dependent variables in the set of equations of the first order is a question of some interest which will now be examined. Let us first consider the equations du __dv du _ dv dx~~dy' dy^fa' ...... W which lead to the equation In this case the quantity w = u + v satisfies a linear equation of the first order ~ ~ cw _ dw fa^dy' and this equation possesses the primary solution w = F (x -f y) which is also a primary solution of the equation (F). Similarly the quantity z = u — v satisfies the equation which possesses a primary solution z = G (x — y) which is also a primary solution of the equation (F). To generalise this result we consider a set of m linear partial differential equations of the first order, L^UI + Ll2u2 -f ... Llmum - 0, j L21U} -f L22uz -f ... L2mum = 0, 1 LmlUi + Lm2u2 + ... Lmmum = Oj where Lvq denotes a linear operator of type (P> V> l) A + (P> V> 2) D2 + ... (p, q, m) Dm, where the coefficients (p, q, r) are constants. Multiplying these equations by coefficients bl9 b2) ... bm respectively, the resulting equation is of the form L (a^ -f a2u2 + ... amum) = 0 ...... (H) 104 The Classical Equations if the constants blt b2, ... 6m; al9 a2, ... am are of such a nature that 4- a) and the operator L is of the form where the operator coefficients Zl5 £2, ... /w are constants to be determined. Equating the coefficients of the operator D in the identities (I) we obtain _,, , v 7 S6p(p, ?;r) = aaJf. This equation indicates that if z1)z2, ... zm; yl , t/2 > - • • 2/m are arbitrary quantities, the bilinear form can be resolved into linear factors When the coefficients bly ... bm can be chosen so that the bilinear form breaks up in this way the two factors will give the required coefficients al9 ... am; 119 ... lm and the partial differential equations will give an ex- pression for (Zji/j -f ... amum which may be called a primary solution of the set of linear partial differential equations of grade m ~ 1. When such a solution exists the system is said to be reducible. The problem of finding when a set of equations is reducible is thus reduced to an algebraic problem. Now let f2 denote the determinant Anl An2 ••• Anro and let An , ... AlTO denote the co-factors of the constituents Ln, L12, ... Llm respectively. If we write it is easily seen that the last m — 1 equations of the set are all formally satisfied, and since the first equation is formally satisfied if v is a solution of the partial differential equation which is of order m. Since £}&! = flAuv = Au£lv — 0, the quantities Ui,u2, ... um are all solutions of the same partial differential equation. = 0, Eeducibility of Equations 105 It should be noticed that a^ + ... amum = (a1An + a2A12 + ... amAlm) v, consequently L (fl^Wj + ... amum) = L (a1An -f a2A12 + ... amAlm) v. The equation £ (a^ + ... amum) = 0 will be a consequence of the equation £lv — 0 if the operator Cl breaks up into two factors L and (axAn + a2A12+ ... amAlm) of which one, L, is linear. The set of linear equations is thus reducible when the equation £lv = 0 is reducible. It is clear from this result that we cannot generally expect a set of linear homogeneous equations of type (G) to possess primary solutions of grade m — 1. The equations do, however, generally possess primary solutions of grade 1. To see this we try *i=/i(0), «2=/2(0), - *m=/*(0). Substituting in the set of equations we obtain the set of linear equations // (0) Ln0 + /2' (0) L120 + ... fm' (0) Llm6 = 0, //'(») V +/•' (0) L220 + .../m' (0) L2m0 = 0, from which the quantities// (0), f2 (0), ... fm' (0) may be eliminated. The resulting equation, T a T a T a LnU L12V ... Llmu L210 L220 ... L2m0 . Lml0 Lm20 ... Lmm0 is no other than the partial differential equation of the characteristics of the equation £lu = 0. § 1-93. Primary solutions of the second grade. We have already seen that the wave-equation possesses primary solutions of type F (0y <f>) which may be called primary solutions of the second grade. The result already obtained may be generalised by saying that if 10, m0, n0, p0, 119 ml9 nl9 pl are quantities independent of #, t/, z and t and connected by the relations the quantities are such that the function u = F (0, <f>) is a solution of Q2u = 0. 106 The Classical Equations This result may be generalised still further by making the coefficients Z0, m0, etc., functions of two parameters a, r and forming the double con- tour integral tt=_ Jiff _ f(a,T)dadT _ 47T2JJ (I0x _ (I0x + m0y + nQz - pQct - g0) (^x -f where/ (a, T), gr0 (a, r), ft (a, r) are arbitrary functions of their arguments. This integral will generally be a solution of the wave-equation and the value of the integral which is suggested by the theory of the residues of double integrals is T , . . n. * u = J-*f(a,P), in which a, /? satisfy M«>j8) = 0, Ao(a,j8) = 0, ...... (K) where A! (a, r) - #/! (a, r) + yml (a, r) + zn^ (a, r) - Ctp1 (a, r) - ft (a, T), A0 (<r, r) = xl0 (a, r) + ymQ (a, r) -f 2W0 (a, r) - Ctp0 (a, r) - gr0 (a, r), and J is the value when cr = a, r = ^8 of the Jacobian This result, which may be extended to any linear equation with a two- parameter family of primary solutions of the second grade, will now -be verified for the case of the wave-equation. It should be remarked that the method gives us a solution of the wave-equation of type where y is a particular solution of the wave-equation. Such a solution will be called a primitive solution ; it is easily verified that the parameters a and j3 occurring in a primitive solution are such that the function v = F (a, j8) is a solution of the partial differential equation of the characteristics Instead of considering the wave-equation it is more advantageous to consider the set of partial differential equations comprised in the vector equations . « ^ =0 ...... (M) and to look for a primitive solution of these equations of type in which / is an arbitrary function of the two parameters a and /?, which are certain functions of x, y, z and t, and the vector q is a particular solution of the set of equations. Substituting in the equations (L) we find that since /is arbitrary a and ]8 must satisfy the equations in o>, cVco x q = — iqda/dt, <?.Vaj = 0, Primitive Solution of Maxwell's Equations 107 which indicate that ^ 3 («.£)_** 8 («,£) -Kd(y,z)~ c 3(x,t)' _ a (a, )8) uc 9 (a. )8) "3(2,*) c8(y, 0' 8 «» *« 9 (<*. (N) where K is some multiplier. To solve these equations we take a, /?, #, y as new independent variables and write the equation connecting a and ft in the form « , ~ JX • ^ / o ^ 8 (g, ft, x, t) = * 8 (a, ft, y, g) 3 (y, z, x,t) cd (x, t, y, z) ' _ (z, x, y,t) c d (y, t, z, x) ' _ ^ 9 (a:, y, g, «) c 3 (z, t, x, y) ' Now multiply each of the Jacobians by ~-?J-%-*— \ and make use of *J J a(a,j3,»,y) the multiplication theorem for Jacobians. We then obtain a set of equations similar to the above but with 3 (a, /J, x, y) in each denominator. The new equations reduce to the form dt __ i dz dt _ i dz d (z, t) _ i dy cdx' dx ~~ c dy ' 8 (x, y) c ' The first two of these equations are analogous to the equations con- necting conjugate functions t and ig/c, consequently we may write z-ct = &[x+ iy,a, /?], z + ct = g [x — 4y* a, j8] . Substituting in the third equation, we find that $'£' = - 1, where in each case the prime denotes a derivative with respect to the first argument. Evidently &' must be independent of x -f iy and <£' inde- pendent of x — iy. The general solution is thus determined by equations of the form 8 _ d _ ^ (a, 0 + (jr + iy) d («, ft, Z + ct = 4,(a,p)-(x- iy) [0 (a, /3)]-1, where 0, <j>, ifi are arbitrary functions of a and /9 which are continuous (Z), 1) in some domain of the complex variables a and ft. For some purposes it is more convenient to write the equations in the equivalent form , , / rt. / . v n , /,, H z - ct = <£ (a, j8) + (x + ty) 6 (a, /5), 0 (a, ft (z + c<) = X («, 0) - (* - *»• 108 The Classical Equations These equations are easily seen to be of the type (K) and may indeed be regarded as a canonical form of (K). When the expressions for q are substituted in the equations (M) it is easily seen that K is a function of a and ft. Since Q already contains an arbitrary function of a and ft we may without loss of generality take K = 1. A case of particular interest arises when <£ = £ (a) - cr (a) - [f (a) + iy (a)] 0, 0 = £ (a) + cr (a) + [£ (a) - irj (a)] 6~\ where f (a), 77 (a), £ («) and r («) are real arbitrary functions of a which are continuous (D, 2). We then have ^TI^"T[^^i^)j ^ ~ r~~ Ti^+7ir-~T(a)] ' and so a is defined by the equation We may without loss of much generality take r (a) = a and use r as variable in place of a. Let us now regard £ (T), 77 (T), £ (T) as the co- ordinates of a point S moving with velocity v which is a function of r. For the sake of simplicity we shall suppose that for each value of r we have the inequality v2 < c2, which means that the velocity of S is always less than the velocity of light. We shall further introduce the inequality T < t. This is done to make the value of r associated with a given space- time point (x, y, z, t) unique*. To prove that it is unique we describe a sphere of radius c (t — r) with its centre at the point occupied by S at the instant r. As r varies we obtain a family of spheres ranging from the point sphere corresponding to r = t to a sphere of infinite radius corresponding to r = — oo. Now, since v* < c2 it is easily seen that no two spheres intersect. Each sphere is, in fact, completely surrounded by all the spheres that correspond to earlier times r. There is consequently only one sphere through each point of space and so the value of r corresponding to (x, y, z, t) is unique. The corresponding position of 8 may be called the effective position of S relative to (x, y, z, t). In calculating the Jacobians r may be treated as constant in the differentiations of ft. Now * Proofs of this theorem have been given by A. Li6nard, L'&lairage tiectrique, t. xvi, pp. 5, i ^, 106 (1898); A. W. Con way, Proc. London Math. Soc. (2), vol. I (1903); G. A. Schott, Electromagnetic Radiation (Cambridge, 1912). Primitive Solution of the Wave-Equation 109 where M=[x-£ (r)] £'(T) + (y ~ V (T)] VW + [Z - I (r)] £'(T) - fa (' - T), and primes denote derivatives with respect to T. We thus find that '§(y, 2)=~2J/(1~^)> a («, 0) »/s a(3,y) JIT The ratios of the Jacobians thus depend only on ft and we have the general result that the function M-I f / R\ is a solution of the wave-equation. When the point (£, T?, £) is stationary and at the origin of co-ordinates this result tells us that if,/ is an arbitrary function which is continuous (/), 2) in some domain of the variables a, ft and if r2 = x* + ?/2 -I- z2 the function , z - r * + ?y is a solution of the wave-equation. There is a corresponding primitive solution of type obtained by changing the sign of t and using another arbitrary function. In the case of the wave-function M~lf (a, /J) the parameter a may be called a phase-parameter because it determines the phase of a disturbance which reaches the point (x, y, z) at time t when the function / is periodic in a. The parameter ft is on the other hand a ray-parameter because a given complex value of /? determines the direction of a ray when a is given. It is easily deduced from the equations (N) that « and /3 satisfy the differential equation of the characteristics , and that + It follows that the quantity v = F (a, ft) is also a solution of (L). An interesting property of this equation (0) is that if a is any solution and we depart from the space-time point (x, y, z, t) in a direction and velocity defined by the equations dx dy dz -r]« sav _ = _ = —- = — - -- as, say, da ca da da dx dy dz dt 110 The Classical Equations then a and its first derivatives are unaltered in value as we follow the moving point. We have in fact , da 7 da , , da , da 7j da=^dx+*ray+~ dz + ^ at ox oy * oz ot 2 2 3% , 32a , r 32« + 3—3" ^ + [3a 32 3i fa 32« 3« 32« 3a 32a 1 da , 3~z 3iaz " c2 - 0. Also, if a and /J are connected by an equation of type (P), 3a 36 1 3a 3B\ 7 , .i_ „ — ^_ . " ) /7o dz dz c2 dt dtj = 0. The equations (0 and P) thus indicate that the path of the particle which moves in accordance with these equations is a straight line described with uniform velocity c and is, moreover, a ray for which j3 is constant. § 1*94. Primitive solutions of Laplace's equation. As a particular case of the above theorem we have the result that the function r \x 4- iy/ is a primitive solution of Laplace's equation. This is not the only type of primitive solution, for the following theorem has been proved*. In order that Laplace's equation may be satisfied by an expression of the form V = yf (0), in which the function / is arbitrary, the quantity 6 must either be defined by an equation of the form [^ - £ (6)1* +(y~r, (*)]» + [z - C (0)? = o, or by an equation of the form xl (6) + ym (6) + zn (6) = p (6), where /, m, n are either constants or functions of 6 connected by the re- lation P + m» + n«-0. The most general value of y is in each case of the form y = no (6) + y*b (6), * See my Differential Equations, p. 202. Primitive Solutions of Laplace's Equation 111 where yx and y2 are particular values of y, whose ratio is not simply a function of 9. In the first case we may take n = w-t, 72 = %-*, where u; = [a; - £ (0)] A (0) + [y - rj (6)] ^ (0) + [2 - £ (0) ] „ (0), t*i = [s - £ (#)] \ (0) + [y - rj (6)] ^ (6) + [z - £ (6)] v, (6), and A, ^, v, Ax, /zx, vl are two independent sets of three functions of 6 which satisfy relations of type A2 + ^ + v2 = 0, A (0) r (0) + p, (0) r)' (6) + v (6) £' (0) = 0. In the second case we may take yl ~ 1 and define y2 by the equation yri = ^ (fl) + ym' (9) + zri (9) - p' (9). If in the first theorem we choose £ = 0, rj — i9, £ ~ 0, we have r2 9=- ----- .--, A-f iu= 0, v = 0, x + ly ^ iv = x + iy, and the theorem tells us that the function is a primitive solution of Laplace's equation. If we write x + iy = t, x — iy = 4:8 this theorem tells us that the function F=r*/(4* + z2/*) is a primitive solution of the equation d*V _ dz* ~ § 1-95. Fundamental solutions*. The equations with primary and primitive solutions have been called basic because it is believed that solutions of a differential equation with the same characteristics as a basic equation can be derived from solutions of the basic equation by some process of integration or summation in which singularities of these solutions of the basic equation fill the whole of the domain under consideration. This point will be illustrated by a consideration of Laplace's equation as our basic equation. We have seen that there is a primitive solution of type r * \x + By a suitable choice of the function / we obtain a primitive solution * These are also called elementary solutions. See Hadamard, Propagation des Ondes. 112 The Classical Equations with singularities at isolated points and along isolated straight lines issuing from isolated singular points. The particular solution F= 1/r has the single isolated point singularity x = 0, y = 0, z = 0. Let us take this particular solution as the starting-point and generalise it by forming a volume integral over a portion of space which we shall call the domain &). When the point (x, y, z) is in the domain I/1 this integral is riot a solution of Laplace's equation but is generally a solution of the equation V2 V + *vF(x, y, 3) = 0, ...... (B) provided suitable limitations are imposed upon the function F. Now the function F is at our disposal and in most cases it can be chosen so as to represent the terms which make the given differential equation differ from the basic equation of Laplace. It is true that this choice of F does not give us a formula for the solution of the given equation but gives us instead an integro-differential equation for the determination of the solution. Yet the point is that when this equation has been solved the desired solution is expressed by means of the formula (A) in terms of primitive solutions of the basic equation. A solution of the basic equation which gives by means of an integral a solution of the corresponding equation, such as (B), in which the additional term is an arbitrary function of the independent variables, is called a fundamental solution. Rules for finding fundamental solutions have been given by Fredholm and Zeilon. In some cases the solution which is called fundamental seems to be unique and the theory is simple. In other cases difficulties arise. In any case much depends upon the domain W and the supplementary conditions that are imposed upon the solution. When the basic equation is the wave-equation the question of a funda- mental solution is particularly interesting. There are, indeed, two solutions, J7 1 I 1 1 and F= - .-}- 2r [r-ct ' r + ct] r* - cW which may be regarded as natural generalisations of the fundamental solution 1/r of Laplace's equation. The former seems to be the most useful as is shown by a famous theorem due to Kirchhoff . In the case of the equation of the conduction of heat the solution which is regarded as fundamental is Fundamental Solutions 113 when the equation is taken in the form _ X* and is V = t~^ e *** when the equation is taken in the simpler form The equation of heat conduction is not a basic equation but may be transformed into a basic equation by the introduction of an auxiliary variable in a manner already mentioned. Thus the basic equation derived from 97 w is =-3- = -s-9- , W = 3s 3J dx2 and this equation possesses the primitive solution of which TF = 2~i exp — - r \_K 4:Ktj is a special case. The theory of fundamental solutions is evidently closely connected with the theory of primitive solutions but some principles are needed to guide us in the choice of the particular primitive solution which is to be regarded as fundamental. The necessary principles are given by some general theorems relating to the transformation of integrals which are forms or developments of the well-known theorems of Green and Gauss. These theorems will be discussed in Chapter II. An entirely different discussion of the fundamental solutions of partial differential equations with constant coefficients has been given recently by G. Herglotz, Leipzig er Berichte, vol. LXXVIII, pp. 93, 287 (1926) with references to the literature. EXAMPLES 1. Prove that the equation -. = ^-j ot dxr is satisfied by the two definite integrals V - 4 f °° e-** (cos xs - sin xs) e'*t8' da, /GO F=*/ v(a,t)v(x,8)ds, where v (x, t) Show also that the two integrals represent the same solution. 114 The Classical Equations 2. Prove that this solution can be expanded in the form V - F0 - F, + F», where F0 = r (i) (4<F l [l - 4, fj + ^ (5)" + •••) 3. Show also that y = 4 / e~4<*4 cos sx cosh sxds, 7o Vl — 4 I e~*i8* [sin sx cosh sx + cos sx sinh 50;] e&, Fo == 4 I e~4<s4 sin sx sinh so; . ds. 2 Jo 4. Prove that there is a fourth solution V3 - x?1rl oi ~ 7 i T "^" n i"\l ( < ) ~ "• " ^ / e~4<s4 tsm 5a: C08^ 5a: — cos sx smn 5a:] ^5- 5. If V (x, t) is a solution of the equation 3V __ d*V ~df " to* * ~ lj the quantity is generally a solution of the set of equations In particular, if s =. 2 and V (x, t) is the function v (x, t) of Ex. 1, the corresponding function yn (t) is yn (0 This may be called the fundamental solution, and when the second form is adopted s may have any positive integral value. In particular, when 5 = 4, this function is de- rivable from t^e function v (x, t) of Ex. 1, p. 113. CHAPTER II APPLICATIONS OF THE INTEGRAL THEOREMS OF GAUSS AND STOKES . In the following investigations much use will be made of the well-known formulae ...... (A) for the transformation of line and surface integrals into surface and volume integrals respectively. In these equations Z, m, n are the direction cosines of the normal to the surface element dS, the normal being drawn in a direction away from the region over which the volume integral is taken or in a direction which is associated with the direction of integration round the closed curve C by the right-handed screw rule. The functions u, v, w, X, Y, Z occurring in these equations will be supposed to be continuous over the domains under consideration and to possess continuous first derivatives of the types required* . The equations may be given various vector forms, the simplest being those in which u, v, w are regarded as the components of a vector q and X, Y, Z the components of a vector F. The equations are then I q . ds = I (curl q) . dS, j _ J J F. dS = [(div F) dr (dr = dx . dy . dz), —^ where ds now stands for a vector of magnitude ds and the direction of the tangent to the curve C, while dS represents a vector of magnitude dS and the direction of the outward-drawn normal. The dot is used to indicate a scalar product of two vectors. Another convenient notation is \qtds = I (curlq)ndS, J ~ ...... (C) / FndS f(divF)dr, * See for instance Goursat-Hedrick, Mathematical Analysis, vol. I, pp. 262, 309. Some in- teresting remarks relating to the proofs of the theorems will be found in a paper by J. Carr, Ph il. Mag. (7), vol. iv, p. 449 (1927). The first theorem is well discussed by W. H. Young, Proc. London Math. Soc. (2), vol. xxiv, p. 21 (1926); and by O. D. Kellogg, Foundations of Potential Theory, Springer, Berlin (1929), ch. iv. 8-2 116 Applications of the Integral Theorems of Gauss and Stokes where the suffixes t and n are used to denote components in the direction of the tangent and normal respectively. If we write Z - v, Y = - w, X = 0; X = w, Z = - u, Y = 0; 7 = ?/, X = — v, Z = 0 in succession we obtain three equations which may be written in the vector form I (qx dS) = - j(curlq)dr, (D) where the symbol x is used to denote a vector product. Again, if we write successively X = p, Y = Z = 0; Y = p, Z = X = 0; Z = p, X = Y = 0, we obtain three equations which may be written in the vector form ^ ...... (E) where Vp denotes the vector with components ^ , ^ ^ , ~ respectively. § 2-12. To obtain physical interpretations of these equations we shall first of all regard u, v, w as the component velocities of a particle of fluid which happens to be at the point (#, y, z) at time t. The quantities £, 77, £ defined by the equations . _ dw dv _du dw ^ _ Sv du *=dy~dz' r]~dz~dx9 ^^dx^dy may then be regarded as the components of the vorticity. The line integral in (A) is called the circulation round the closed curve C and the theorem tells us that this is equal to the surface integral of the normal component of the vorticity. When there is a velocity potential </> we have ~ , - , ~ , dd> O(h o<p u = -f- , v = -£- , w = /- dx dy dz (in vector notation q — Vcf>) and £ = y = £ = 0, the circulation round a closed curve is thenxzero so long as the conditions for the transformation of the line integral into a surface integral are fulfilled. The circulation is not zero when . , . , , x </> = tan-1 (y/x), and the curve C is a simple closed curve through which the axis of z passes once without any intersection. The axis of z is then a line of singularities for the functions u and v. The value of the integral is 27r, for ^ increases by 2n in one circuit round the axis of z. The velocity potential </> = (r/27r) tan-* (y/x) may be regarded as that of a simple line vortex along the axis of z, the strength of the vortex being represented by the quantity F which is- supposed to be constant. F represents the circulation round a closed curve which goes once round the line vortex. Equation of Continuity 1F7 If we write X = pu, Y = pv, Z = pw, where p is the density of the fluid, the surface integral in (A) may be interpreted as the rate at which the mass of the fluid within the closed surface S is decreasing on account of the flow across the surface 8. If fluid is neither created nor destroyed within the surface this decrease of mass is also represented by The two expressions are equal when the following equation is satisfied at each place (#, y, z) and at each time t, This is the equation of continuity of hydrodynamics. There is a similar equation in the theory of electricity when p is interpreted as the density of electricity and u, v, w as the component velocities of the electricity which happens to be at the point (#, ?/, z) at time t. When p is constant the equation of continuity takes the simple form du dv div __ dx + dy + dz = ° (in vector notation div q = 0). This simple form may be used also when dp/dt = 0, where d/dt stands for the hydrodynamical operator d a a a a i4 = ^ + u n + v ~T + w a~ » «£ cM ra cty dz a fluid for which rfp/Y/£ = 0 is said to be incompressible. When p is interpreted as fluid pressure the equation (E) indicates that as far as the components of the total force are concerned the effect of fluid pressure on a surface is the same as that of a body force which acts at the point (x9 y, z) and is represented in magnitude and direction by the vector — Vp, the sign being negative because the force acts inwards and not outwards relative to each surface element. Putting q = pr in equation (D), where r is the vector with components x, y, z, we have an equation I (r x pds) = — I (curler) dr = (r x Vp) dr, which indicates that the foregoing distribution of body force gives the same moments about the three axes of co-ordinates as the set of forces arising from the pressures on the surface S. The body forces are thus completely equivalent to the forces arising from the pressures on the surface elements. This result is useful for the formulation of the equations of hydrodynamics which are usually understood to mean that the mass multiplied by the acceleration of each fluid element is equal to the total body force. If in addition to the body force arising from the pressure there is a body force F whose components per unit mass are JL , 7, Z for a particle 118 Applications of the Integral Theorems of Gauss and Stokes which is at (x, y, z) at time t, the equations of hydrodynamics may be written in the vector form When viscosity and turbulence are neglected the body force often can be derived from a potential fi so that F = VQ. The hydrodynamical equations then take the simple form , which implies that in this case there is an acceleration potential if p is a constant or a function of p. When in addition there is a velocity potential <f> the equations may be written in the form and imply that f dp~ + ?J + |92 = Q + / (t), J p 01 where/ (t) is some function of t. This may be regarded as an equation for the pressure, when ti = 0 it indicates that the pressure is low where the velocity is high. § 2-13. The equation of the conduction of heat. When different parts of a body are at different temperatures, energy in the form of heat flows from the hotter parts to the colder and a state of equilibrium is gradually established in which the temperature is uniformly constant throughout the body, if the different parts of the body are relatively at rest and do not participate in an unequal manner in heat exchanges with other bodies. When, however, a steady supply of heat is maintained at some place in the body, the steady state which is gradually approached may be one in which the temperature varies from point to point but remains constant at each point. A hot body is not like a pendulum swinging in air and performing a series of damped oscillations as the position of equilibrium is approached, it is more like a pendulum moving in a very viscous fluid and approaching its position of equilibrium from one side only. The steady state appears, in fact, to be approached without oscillation. These remarks apply, of course, to the phenomenon of conduction of heat when there is no relative motion (on a large scale) of different parts of the body. When a liquid is heated, a state of uniform temperature is produced largely by convection currents in which part of the fluid Amoves from one place to another and carries heat with it. There are convection currents also in the atmosphere and these are responsible not only for the diffusion of heat and water vapour but also for a transportation of momentum which is responsible for the diurnal variation of wind velocity and other phenomena. Conduction of Heat 119 A third process by which heat may be lost or gained by a body is by the emission or absorption of radiation. This process will be treated here as a surface phenomenon so that the laws of emission and absorption are expressed as boundary conditions ; the propagation of the radiation in the intervening space between two bodies or between different parts of the same body is considered in electromagnetic theory. The mechanism of the emission or absorption is not fully understood and is best described by means of the quantum theory and the theory of the electron. The use of a simple boundary condition avoids all the difficulty and is sufficiently accurate for most mathematical investigations. In many problems, how- ever, radiation need not be taken into consideration at all. The fundamental hypothesis on which the mathematical theory of the conduction of heat is based is that the rate of transfer of heat across a small element dS of a surface of constant temperature (i.e. an isothermal surface) is represented by ™ where K is the thermal conductivity of the substance, 0 is the temperature in the neighbourhood of dS, and ~~ denotes a differentiation along the outward-drawn normal to dS. The negative sign in this expression simply expresses the fact that the flow of heat is from places of higher to places of lower temperature. The rate of transfer of heat across any surface element dv in time dt may be denoted by fvdadt, where the quantity fv is called the flux of heat across the element and the suffix v is used to indicate the direction of the normal to the element. Let us now consider a small tetrahedron DABC whose faces DBC, DC A, DAB, ABC are normal respectively to the directions Ox, Oy, Oz, Ow, where the first three lines are parallel to the axes of co-ordinates. Denoting the area ABC by A, the areas DBC, DC A, DAB are respectively wx&, Wy&, wzA, where wx, wy, wz are the direction cosines of Ow. Wher A is very small the rate at which heat is being gained by the tetrahedron at time t is approximately (wxfx + wvfy + wjz - /„) A. 7/J This must be equal to. Vcp ^~ , where V is the volume of the tetrahedron, u/t c the specific heat of the material and p its density. Now V = £jpA, where p is the perpendicular distance of D from the plane ABC, hence Wxfx + Wyfv + Wzfz - /« = $PCP ^ and so tends to zero as p tends to zero. When DAB is an element of an isothermal surface we may use the 120 Applications of the Integral Theorems of Gauss and Stokes additional hypothesis that fx and fy are both zero and the equation giV6S f f L, -= wz j z = Jw zjz The law (A) thus holds not simply for an isothermal surface but for any surface separating two portions of the same material. The vector A0 whose components are x- - , ~ , ~ is called the temperature gradient at the point (x, y, z) at time t. Let us now consider a portion of the body bounded by a closed surface R. Assuming that fff , fy , fz and their partial derivatives with respect to x, y and z are continuous functions of x, y and z for all points of the region bounded by /V, the rate at which this region is gaining heat on account of the fluxes across its surface elements is Transforming this into a volume integral and equating the result to J7/3 cp ' dxdydz, (IT we have the equation ! \CP n i — -5~ ( & * ' " -r K ^~ > ' — ~>- ( ^ ^ } \ dxdydz. ]jj[rdt 3x{ 3x 3y 3y 3z\ 3z J y This must hold for any portion of the material that is bounded by a simple closed surface and this condition is satisfied if at each point cp (T ._ div (KV0) = 0. a I If the body is at rest we can write ^ in place of -j- , but if it is a moving ut (Jit/ Jf\ fluid the appropriate expression for -7- is d9 30 30 30 30 ~n ^ bi 'I ^ ^ f- V >r -h W x- , dt dt ox cy 3z where u, v and w are the component velocities of the medium. In most mathematical investigations the medium is stationary and the quantities c, K and p are constant in both space and time and the equation takes the simple form ^ dt in which K is a constant called the diffusivity*. If at the point (x, y, z) there is a source of heat supplying in time dt a quantity F (x, y, z,t) dxdydzdt * This name was suggested by Lord Kelvin. A useful table of the quantities K, c, p and x is given in Ingersoll and Zobel's Mathematical Theory of Heat Conduction (Ginn & Co., 1913). The Drying of Wood 121 of heat to the volume element dxdydz, a term F (x, ?/, z, t) must be added to the right-hand side of the equation. A similar equation occurs in the theory of diffusion ; it is only necessary to replace temperature by concentration of the diffusing substance in order to obtain the derivation of the equation of diffusion. The quantity of diffusing substance conducted from place to place now corresponds to the amount of heat that is being conducted. The theory of diffusion of heat was developed by Fourier, that of a substance by Fick. In recent times a theory of non-Fickian diffusion has been developed in which the coefficient K is not a constant. Reference may be made to the work of L. F. Richard- son*. § 2-14. An equation similar to the equation of the conduction of heat has been used recently by Tuttle f in a theory of the drying of wood. It is known that when different parts of a piece of wood are at different moisture contents, moisture transfuses from the wetter to the drier regions ; Tuttle therefore adopts the fundamental hypothesis that the rate at which transfusion takes place transversely with respect to the wood fibres or elements is proportional to the slope of the moisture gradient. This assumption leads to the equation d_e dW di W where 9 is moisture content expressed as a percentage of the oven-dry weight of the wood and It2 is a constant for the particular wood and may be called the transfusivity (across the grain) of the species of wood under consideration. From actual data on the distribution of moisture in the heartwood of a piece of Sitka spruce after five hours' drying at a temperature of 160° F. and in air with a relative humidity of 30 %, Tuttle finds by a computation that h2 is about 0-0053, where lengths are measured in inches, time in hours and moisture content in percentage of dry weight of wood. The actual boundary conditions considered in the computation were 6 = 0 at x - 0, 9 = 0 at x - 1, 0 = 00 when t - 0. A more complete theory of drying has been given recently by E. E. LibmanJ in his theory of porous flow. He denotes the mass of fluid per unit mass of dry material by v and calls it the moisture density. The symbols p, a, r are used to denote the densities of moist material, dry material and fluid respectively and ft is used to denote the coefficient of compressibility of the moist material. * Proc. Roy. Soc. London, vol. ex, p. 709 (1926). f F. Tuttle, Journ. of the Franklin Inst. vol. cc, p. 609 ( 1925). t E. E. Libman, Phil. Mttg. (7), vol. iv, p. 1285 (1927). 122 Applications of the Integral Theorems of Gauss and Stokes The rate of gain of fluid per unit mass of dry material in the volume V is the rate of increase of v, where v is the average value of v in F. If w is the mass of dry material in volume V of moist material and fn = mass of fluid flowing in unit time across unit area normal to the direction n we have dv V therefore ~-. = --- div/. ...... (B) ot w • Now, the mass of fluid in the volume V is wv and the total mass of material in V is wv -j- w and is also pF, hence and (B) gives the equation < 'dv I + v 3t=~ >" for the interior of the porous body. If EdS denotes the mass of fluid evaporating in unit time from a small area dS of the boundary of the porous body the boundary condition is fn = E. The flow of fluid in a porous material may be regarded as the sum of three separate flows due respectively to capillarity, gravity and a pressure gradient caused by shrinkage. We therefore write, for the case in which the z axis is vertical and p is the pressure, - „ dv j dz 7 dp /.--Kfr-kgrfc-kft, where K and k are constants characteristic of the material. \-\-v Consider now a small element of volume — — 8w at the point P (#, y, z), the associated mass of dry material being 8w and the volume per unit mass of dry material 1 + v Then "- = " dv dv dp dp dV dp d /I -j- v\ fo~WdvssdVdv(~j~)' 1 dV But by definition B = — ^7 ~T- , K ap therefore $=- l d f1 + ^- ! d /-- X + v — - -j- _ Vfidv\ p 1 rf A 1 + \ 1 d A 1 -f v\ ) = _ - (w — IU . ) ) fidv\ 6 p /' The Heating of a Porous Body 123 ™ ^- d<f> „ Putting dl-X- we have ..- or /• - - f -- f - - kar Jx~ dx' Jv~ dy' h~ c!z~ J ' div/= - V*<f>, and so tp ^ = W, 1 + t> 3< r wliile the boundary condition takes the form *- + IS + tgr* = 0. It should be mentioned that in the derivation of this equation the material has been assumed to be isotropic. In the special case of no shrinkage we have p = a (1 -f v), ,- = K , cf> = JSTi; 4- const., and the equation for v becomes which is similar in form to the equation of the conduction of heat. The boundary condition is ^ ~ ,r ov n 7 oz A" + E 4- kgr - 0. dn on § 2-15. The heating of a porous body by a warm fluid*. A warm fluid carrying heat is supposed to flow with constant velocity into a tube which contains a porous substance such as a solid body in a finely divided state. For convenience we shall call the fluid steam and the porous substance iron. The steam is initially at a constant temperature which is higher than that of the iron. The problem is to determine the temperatures of the iron and steam at a given time and position on the assumption that the specific heats of the iron and steam are both constant and that there are no* heat exchanges between the wall of the tube and either the iron or steam, no heat exchanges between different particles of steam and no heat exchanges between different particles of iron. The problem is, of course, idealised by these simplifying assumptions. We make the further assumption that the velocity of the steam is the same all over the cross-section of the pipe. This, too, would not be quite true in actual practice. Let U be the temperature of the iron at a place specified by a co-ordinate x measured parallel to the axis of the pipe, V the corresponding temperature * A. Anzelius, Zeit*. f. ang. Math. u. Mech. Bd. vi, S. 291 (1926). 124 Applications of the Integral Theorems of Gauss and Stokes of the steam. These quantities will be regarded as functions of x and I only. This is approximately true if the pipe is of uniform section so that the cross-sectional area is a constant quantity A. Let us now consider a slice of the pipe bounded by the wall and two transverse planes x and x f- dx. At times t and t -f dt the heat contents of the iron contained in this slice are respectively uUA dx and u ( U + ~ dt] A dx, V at J where u is the quantity of heat necessary to raise the temperature of unit volume of the iron through unit temperature. Thus the quantity of heat imparted to the iron in the slice in time dt is dQ,= uA~dtdx. ct Similarly, at time t the heat content of the vapour in the slice is rVA dx ( dV \ and at time / h dt it is v ( V -f -^ - dt] A dx, where v is a quantity analogous to u. With the steam flowing across the plane x in time dt a quantity of heat vVAcdt is brought into the slice where c is the constant velocity of flow. In the same time a quantity of heat v[V+ -— dx]Acdt leaves the slice across V ox / the plane x -f <tx. The steam has thus conveyed to the iron a quantity of heat ~ir - Tr i^ (3V 5V\ A 7 7 dQ2 = -- v -,- -I- c ~ }A dtdx. \ ot ox/ In accordance with the law of heat transfer that is usually adopted the quantity of heat transferred from the steam to the iron in the slice in time dils d#3 = k(V- U) A dtdx, where k is the heat transfer factor for iron and steam. We thus have the equations With the notation a = k/cv, b = k/cu and the new variables £ = ax, r = b (ct — x), the equations become W-u-v du-v-u 3{~ ' ST y These equations imply that the quantity A (£, r) defined by b(S,T) = #**(V-U) is a solution of the partial differential equation Laplace's Method 125 The supplementary conditions which will be adopted are tf (£0)= U19 7(0, T)= F1? where D^ and Fx are constants. The equation (A) then gives f/(0, r)= ^-(Fi- 1^)6-', F(£,0) = C/x+CFj- tfje-f, and so the supplementary conditions for the quantity A are A (f , 0) = A (0, r) = F! - t/! = IF, say. § 2-16. Solution by the metfwd of Laplace. The equation (A) may be solved by a method of successive approximations by writing A- AO + Ax + A2+ ..., where \ = W and ~ *- = A^^. This gives A - JF/0 [2 V(fr)J, V - U= TFe-<^>/ rax =. Vl - H'c-6 <«*-*> e~s /0 [2 V{^ (c^ - x)}] ds, Jo = Ul + TFe~a^ f C< ^c-'/o [2 J o u = c/i + For x> ct the solution has no physical meaning but for such values of oc the iron has not yet been reached by the steam and so U — C^. As t -> oo we should have U -+ V19 V -> V1 ; this condition is easily seen to be satisfied, for our formula for V — U indicates that V — U -> 0 and U -> Ft because [ e~s 70 [2^/(axs)] ds = eax. Jo The properties of the solution might be used, however, to infer the value of this integral. EXAMPLE Prove that if E (r) = 2 -^-— 3 , the differential equation ^—^—^ = V ^ . dxdydz fV (z is satisfied by V=\ I </>(v, w)E{x(y — v)(z — w)}dvdw [z (x -f I i/i(w,u) E {y(z — w)(x — u)}dwdu [x fy Jo Jo + IXP (u) E {(x - u) yz} du + (VQ(v) E {(y - v) zx} dv J 0 J 0 + [ZE (w) E {(z - w) xy} dw + SE (xyz). [T. W. Chaundy, Proc. London Math. Soc. (2), vol. xxi, p. 214 (1923).] 126 Applications of the Integral Theorems of Gauss and Stokes § 2-21. Riemanris method. Let L (u) be used to denote the differential expression a 02U CU , OU ~ a 4- #0 + 63 + cuy dxdy dx dy where a, b, c are continuous (Z), 1) in a region ^? in the (x, y) plane. The adjoint expression L (v) is defined by the relation where M and N are certain quantities which can be expressed in terms of u, v and their first derivatives. Appropriate forms for L, M and N are S2v d d L (v) = - -- ^ (at;) - ~ (6v) + cv, dxdy 9^; ty nf I / du dv\ 19, v ^ , v If = aw +v-u = (uv) - uP (v), say, 'w 9i; 2 aa. - « , T> 7 x 9v ^ 7 . dv ^ , where P (v) = - - av, Q (v) = x —60;. Now if C' is a closed curve whiclMi^s entirely within the region R and if both u and v are continuous (D, 1) in ,&, we have by the two-dimensional form of Green's theorem (IM + mN) ds^ + dxdy = [vL (u) - ul (v)] dxdy, where /, m are the direction cosines of the normal to the curve C and the double integral is taken over the area bounded by C, and so will be ex- pressed in terms of the values of u and its normal derivative at points of the curve F, for when u is known its tangential derivative is known and ^ - and XT- can be expressed in terms of the normal and tangential de- rivatives. If (XQ, y0) are the co-ordinates of the point A the function v which enables us to solve the foregoing problem may be written in the form , x v = g (x, ?/; x0, #0), and may be called a Green's function of the differential expression L (u). This theorem will now be applied in the case where the curve C consists of lines XA, A Y parallel to the axes of x and y respectively and a curve I1 joining the points Y and X. Using letters instead of particular values of the variable of integration to denote the end points of each integral, we have when L (u) = 0, L (v) = 0, tXNdx- \ J A J Riemann's Method 127 [A [A Now — M dy = 4 \(uv)Y — (uv)A] + uP (v) dy, JY JY f A f x and Ndx = J [(w)x - (uv) 4] - uQ (v) dx, JA J A rx therefore (uv)A = i [(w)v + (wv)r] + (IM +.mN) efe J Y CA rx -f ?/P (v) dy — wQ (v) eu-. J i' .' .4 If now the function v can be chosen so that P (v) = 0 on ^4 7 and Q (v) = 0 on AX, the value of ^ at the point A will be given by the formula rx (uv)A = J .[(uv) v + (^')r] -f (J-W + mN) ds\ i Y It should be noticed that if u is not a solution of L (u) ~ 0 but a solution of the corresponding expression for u is (uv)A = $ [(uv)x + (uv)Y] + I (IM -h mN) ds -f- | U/(.r, ? r An interesting property of the function y may be obtained by consider- ing the case when the curve F consists of a line YB parallel to AX and a line BX parallel to YA. We then have x [A (IM -f mN) ds = \XMdy- {" Ndx, J Y J B J Y also M = - ^ V (wv) + ^^ M, ^ (w) = a^ + N^~l L(uv] + "Q (u]- Q (u] = ^x+ [B [B - we have Ndx = \ [(uv)Y — (uv)E] + vQ (u) dx, J Y J Y J Mdy - | [(uv)B - (uv)x] + f- vP (u) dy. B J B CB „ r Hence (^^)x = (uv)B — vQ (u) dx + JY J Now let a function u = h (x,y\xl, y^) be supposed to exist such that Z/ (w) = 0, P (w) = 0 on JBX, Q (u) = 0 on J?7 ; the co-ordinates a?!, ^ being those of B. The formula then gives (uv)A - (ttt;)^. Choosing the arbitrary constant multipliers which occur in the general expressions for g and h, in such a way that 9 (z0> y0l x0t y0) - 1, h (xlf y^ xl9 yj = 1, 128 Applications of the Integral Theorems of Gauss and Stokes the preceding relation can be written in the form h (x0, y0; xl9 yj = g (xly y^ XQ, yQ). When considered as a function of (x, y) the Green's function g satisfies the adjoint equation L (v) = 0, but when considered as a function of (XQ, yQ) it satisfies the original equation expressed in the variables xQy yQ. EXAMPLE Prove that the Green's function for the differential equation is and obtain Laplace's formula tt= f*/ Jo for a solution which satisfies the conditions .5- = </> (x) when x = 0. oy § 2-22. Solution of the equation ^ Let the curve B consist of a line A1A2 parallel to the axis of x and two curves Cl7 C2 starting from Aly Az respectively and running in an upward direction from the line A1A2. Let S denote the realm bounded by the portion of B below a line y parallel to the axis of x and the portion of B which lies between Cl and (72. When y is replaced by the parallel lines y0, y the corresponding realms will be denoted by SQ and 8' respectively. The portions of B below the lines y, y0 , y will be denoted by /?, /?0 and ft' respectively. The equation of A1A2 will be taken to be y = ylt We shall now suppose that y and y' both lie below y0 and that z is a solution of (A) which is regular in S0, regularity meaning that z, x-*, ~ - OX 01/ a2- and ,5-j are continuous functions of x and y in the realm S0 . OX The differential expression adjoint to L (z) is L (f), where and we have the identity t [L (,) -/(*, „)] - ZL (0 = « - Generalised Equation of Conduction 129 Hence if. L (z) = f (x, y), and L (t) = 0, we have ^ | [« | - **] - | [to] - «/- 0. Let us now write £ = 2 (£, 77),. r = £ (£, T;) and integrate the last equation over the region S', then /, ««- Jr [*« + Mr '*)*•] -//, In this equation we write r (£ ,) = T [*, y; f, 77] EE (y - 77)-* e^p [- (* - g)*/(y - 77)], and we take y to be a line which lies just below the line y which passes through the point (x, y). Our aim now is to find the limiting value of the integral on the left as y -> y. By means of the substitution % = x -+- 2u\/(y — 77) this integral is transformed into /•tta 2 s [a: -f 2tt (y - 17)*, 77] e-wt^. .' MI If the equations of the curves Ol , <72 are respectively ar = cx (y), a; - c2 (y), the limits of the integral are respectively ^i = [Ci (17) - a;]/2 V(y ~ T?), ^2 = [Cz (>?) - x]/2 ^/(y - y). If the point (x, y) lies within S we have ut -> — oo, i^2 -*• + oo as yx -> y and 17 -+ y ; if it lies outside $ we have u± -> w2 -> ± oo as yx -v y and ?} -+ y. Finally, if the point (x, y) lies on either 'C1 or (72 one limit is zero, thus we may have either ut -> 0, u2 -+ oo , or u± -> — oo, w2 -^ 0. The Umiting value obtained by putting rj ~ yin the integral is 2z (x, y)\/7r in the first case, zero in the second case and z (x, y) \Ar in the third. Hence when the limiting value is actually attained we have the formula z (xy y) [2V", 0 or VI = f \T£df + (T § - £ ^) ^1 - f f ^ L \ ^? c/f / J .-Js The transition to the limit has been carefully examined by Levi*, Goursatf and Gevreyf. The last named has imposed further conditions * E. E. Levi, AnnaLi di Matematica (3), vol. xiv, p. 187 (1908). t E. Goursat, TraiU # Analyse, t. in, p. 310. t M. Gevrey, Jvurn. de Mathdmatiques (6), t. ix, p. 305 (1913). See also Wera Lebedeff, Diss. Oottingen (1906); E. Holmgren, Arkiv for Matematik, Astronomi och Fyaik, Bd. in (1907), Bd. iv (1908); G. C. Evans, Amer. Journ. Math. vol. xxxvn, p. 431 (1915). 130 Applications of the Integral Theorems of Gauss and Stokes in order to establish the formula in the case when (x, y) lies on either Cl or C2 . His conditions are that, if (f> (s) is continuous, lim [c, (y) - c, (rj)] (y - ^ = 0 (p = 1, 2), r)->y that f [c, (y) - c, (*)] (y - *)-* <f> (s) ds . exp [- {cp (y) - cp (s)}*/4 (y - s)} -' >) should exist and that the functions q (y), c2 (y) be of bounded variation. It may be remarked that the line integrals in this formula are particular solutions of the equation ~ = ~ 2, while the integral _ I I ? (x, y; / 7T .' „' , _ ^ \/ 7T .' „' >o is a particular solution of the equation (A). Sufficient conditions that this may be true have been given by Levi and less restrictive conditions have been formulated by Gevrey. The properties of the integrals f f dT I(x,y) = T(x)y^)r])(f>(7])dr)) J(x,y) = ~ <f> (77) dy . o • c ^"^ have also been studied, where O is a curve running from a point on the line y = yl to a point on the line y = y. It appears that when the point P crosses the curve C at a point P0 the integral / suffers a discontinuity indicated by the formula lim (JP - JP) - ± <f>Po VTT, P->PO the sign being + or — according as P approaches P0 from the right or the left of the curve C. In this formula </> denotes any continuous function and a suffix P is used to denote the value of a function of position at the point 'P. A Green's function for the region 8 may be defined by the formula G (x, y; €,-n) = T (x, y; f9r))-H (x, y; £ 77), r)%Pf T^T-f where H (x, y; £, ??), which satisfies the equation -^ +-*— = 0, is zero on c£ or) y when considered as a function of f and 77, which is regular and which takes the same values as T (x, y ; £, T?) on the curves Ct and C2 . The function n *• « *u . x- 92# W 3*G , SG A . G satisfies the two equations ^ = - , -x>2- + ^— = 0, is zero when x = cl(y)y when ^ = Cj (77), when x = c2 (y) and when f = c2 (77), and is positive in $. With the aid of this function a formula -ff J J S The Green's Function 131 may be given for a solution of (A) which takes assigned values on /?. The problem of determining O is reduced by Gevrey to the solution of some integral equations. Fundamental solutions of the equations dz _ a3z d*z _ d*z dy dx3' dy* dx* have been obtained by H. Block* and have been used by E. Del Vecchiof to obtain solutions of the equations dz d*z_f d*z d*z _f dy ~dx*~J (X' y)' dy* ^8x*~J (X> y}' EXAMPLES 1. Show by means of the substitutions that the integral ' ' (x - f)*(y - ?)-«exp[- (x - II. has a meaning when p + 1 > 0 and p — 2q + 3 > 0, / being an integrable function . [E. E. Levi.] 2. Show that by means of a transformation of variables x' = x' (x, y), y' = ±y the parabolic eq nation ~ -f a ~ -f 6 - 4- cz -f / = 0 ^ n dx2 dx dy may be reduced to the canonical form a22 dz dz a r /2 — a , = a ^~, + cz +f. dx2 dy dx Show also that the term p , may be removed by making a substitution of type z — uv ox and that the term involving u will disappear at the same time if d'2a da da rt dc a- /2 = a a , 5- , -f 2 . , . dx 2 dx dy dx 3. If a, b and c are continuous functions in a region R a solution of 0 (6<0) dx2 dx dy which is regular in R can have neither a positive maximum nor a negative minimum. Hence show that there is only one solution of the equation which is regular in R and has assigned values at points of a closed curve C lying entirely within R. [M. Gevrey.] § 2-23. Green's theorem for a general linear differential equation of the second order. Let the independent variables xlf x2) ... xm be regarded as rectangular co-ordinates in a space of ra dimensions. The derivatives of * Arkivf. Mat., Ast. och Fysik, vol. vn (1912), vol. vm (1913), vol. ix (1913). t Mem. d. R. Accad. d. Sc. di Torino (2), vol. LXVI (1916). 9-2 132 Applications of the Integral Theorems of Gauss and Stokes a function u with respect to the co-ordinates may be indicated by suffixes ^\ ^2 written outside a bracket, thus (u)2 stands for x and (u)23 for •5—^ — . (7it*2 @&2 ^*^3 We now consider the differential equation L (u) = S S ^rs Mr, + S £r (tt)r -f Fw = 0, >=ls-l r-1 /I _ J •^rs -^sr* where the coefficients Ars, Br, F are functions of xl9 x2, ... xm. The expression Z (v) adjoint to L (u) is L(v)= S 2 (4rsv)rf- S(flrt;)r + JTt;, r-l s»l r-1 and we have the identity vL(u)-uL(v)= S (^r)r, r-l m 77t r m where Qr = - u 2 ^4rs (v\s + v S -4rs (u), 4- ^v £r— S (,4rs The ^-dimensional form of the theorem for transforming a surface integral into a volume integral may be written in the form rr m rr m \ \ V(Qr)r dx, ... dxm=-\\ ZnrQrdS, where nl9 n2, ... nm are the direction cosines of the normal to the hyper- surface S, the normal being drawn into the region of integration. Hence we have the equation f f — r r [vL (u) - uL (v)] dxl ... dxn = — \ \{v Dnu - uDnv — uvPn] dS9 vi m where Dn (u) = £ 2 nsArs (u)r, m r m n in Let us write S nsArs = A^r, where vl9 v2, ... vm are the direction cosines of a line which may be called the conormal*, then Dn (u) — A 2 vr (u)r = A (u)v. r-l § 2-24. The characteristics of a partial differential equation of the second order. Let the values of the first derivatives (u)l9 (u)2, ... (u)m be given at points of the hypersurface 6 (xl9 x2, ... xm) = 0. If dxl9 dx2, ... dxm are increments connected by the equation (9)idxl + (0)2 dx2 4- ... (0)m rfa?w = 0, * This is a term introduced by R. d'Adh^mar. Characteristics 133 and if (0^ ^ 0, we may regard the increments dx2 , dx3 , ... dxm as arbitrary, and since , r/ x n , v , , v , , . 7 = (w^a-rj -f (u)p2dx2 -f ... (M)pm&rm, the quantities (0)! (M)^ — (6)s (u)pl may be regarded as known. Similarly the quantities (*)i WIP - (»)p (")n may be regarded as known and so the quantities may be regarded as known. Substituting the values of (u)ps in the partial differential equation m m in L(u)= S 2 ^4rs (w)r» 4- 2 Br (u)r -f /V = 0, (I) r-l s-1 r-1 we see that we have a linear equation to determine (u)n in which the coefficient of (u)n is A = S S 4r,(0)r(0),. (II) If this quantity is different from zero the equation determines (u)n uniquely, but if the quantity A is zero the equation fails to determine (u)n and the derivatives (u) are likewise not determined. In this case the hypersurface 6 (x11 x2, ... xm) = 0 is called a characteristic and the dif- ferential equation A = 0 is called the partial differential equation of the characteristics. The equations of Cauchy's characteristics for this partial differential equation of the first order are dxi __ dx2 _ dxm and these are called the bicharacteristics of the original partial differential equation. All the bicharacteristics passing through a point (a^0, a:2°, ... xm°) generate a hypersurface or conoid with a singular point at (xf, x2°, ... #m°). When all the quantities A^ are constants this conoid is identical with the characteristic cone which is tangent to all the characteristic hypersurfaces through the point (x-f*, x2°, ... xm°). For the theory of characteristics of equations of higher order reference may be made to papers by Levi* and Sanniaf. These authors have also considered multiple characteristics and Sannia gives a complete classifica- tion of linear partial differential equations in two variables of orders up to 5. * E. E. Levi, Ann. di Mat. (3 a), vol. xvi, p. 161 (1909). t G. Sannia, Mem. d. R. Ace. di Torino (2), vol. LXIV (1914); vol. LXVI (1916) 134 Applications of the Integral Theorems of Gauss and Stokes § 2-25. The classification of partial differential equations of the second order. A partial differential equation with real coefficients is said to be of elliptic type when the quadratic form is always positive except when X1 = X2 = ... == Xn = 0. The use of the words elliptic, hyperbolic and parabolic seems natural in the case n = 2, the term to be used depending upon the nature of the conic AnX^ + 2A12X1X2 + A22X2* = 1. For a non-linear equation F (r, ,9, t, p, q, z) = 0 r _ asz 3*z_ _ 32z _dz I """ a*2' '• "" dxdy' ~ dy*' P ~~ dx' there is a similar classification depending on the nature of the quadratic form dF dF dF \ * _J_ 9 V X 4- Y 2 Vr Al dr +^1^29^-+ As -dt. When n > 2 the classification is not so simple ; for instance, when n — 3 it may be based on the different types of quadric surface and it is known that there are two different types of hyperboloid. The word ellipsoidal might be used in this case instead of elliptic, but it seems better to use the same term for all values of n because the im- portant question from the standpoint of the theory of partial differential equations is whether the equation is or is not of elliptic type. For an equation of elliptic type the characteristics are all imaginary and this fact has a marked influence on the properties of the solutions of the equation. When n = 2 typical equations of the three types are d2u d*u du , du d*u du , du ^ n i i. x ~»~- + a ~ 4- 0 * -f cu — 0 (hyperbolic), dxdy dx dy J± ; d2u du . du . . , T , « -2 -f a - -h o ~ + cu = 0 (parabolic). A notable difference between elliptic and hyperbolic equations arises when a solution is required to assume prescribed values at points of a closed curve and be regular within the curve. For illustration let us con- sider the case when the curve is the circle x2 -f y2 = 1. If the boundary condition is V — sin 2nO when x — cos 0, y = sin 6, where n is a positive ?2K integer, there is no solution of the equation ~ ~ — 0 which is continuous (D, 1) and single- valued within and on the circle*, but there is a regular * When «.— ! there is a solution V = 2y (1 — y2)^ which satisfies the boundary condition but dV its derivative — - is infinite on the circle. cy Classification of Equations 135 927 327 solution of -=- 2 + O-T = °> namely, V = r2n sin 2n0. On the other hand, if the boundary condition is V = sin (2n + 1) 9, there is a solution of 32F * -----= 0 of type V = f(y) which satisfies the conditions and is single- valued and continuous in the circle, but this solution is not unique because V = 1 — x2 — y2 is a solution of ~— ~- = 0 which is zero on the circle and y dxdy single -valued and continuous inside the circle. When the solution of a problem is not unique or when there is some uncertainty regarding the existence of a solution the problem may be regarded as not having been formulated correctly. An important property of the boundary problems of mathematical physics is that the correct formulation of the problem is indicated by the physical requirements in nearly every case. §2-26. A property of equations of elliptic type. Picard*, Bernsteinj and Lich tens tern t have shown that the solutions of certain general differential equations of elliptic type cannot have maximum or minimum values in the interior of a region within which they are regular. This property, which has been known for a long time for the case of Laplace's equation, has been proved recently in the following elementary way§. Let L (u) = T A^ (u)^ 4- S Bv (u)v 1,1 i be a partial differential equation of the second order whose coefficients AHV) Bv are continuous functions of the co-ordinates (xl> x2, ... xn) of a point P of an n-dimensional region T. For convenience we shall sometimes use a symbol such as u (P) to denote a quantity which depends on the co- ordinates of the point P. We can then state the following theorem : // u (P) is continuous (£), 2) and satisfies the inequality L (u) > 0 every- where in T, an inequality of type u (P) < u (P0) can only be satisfied through- out T, where P0 is a fixed internal point, when the inequality reduces to the equality u (P) — u (P0). Similarly, if L (u) < 0 throughout T, the inequality u (P) > u (P0) in T implies that u (P) = u (PQ). The proof will be given for the case n = 2 so that we can use the familiar terminology of plane geometry, but the method is perfectly general. Let us suppose that L (u) > 0 in T and that u (P0) = M , while u (P) < M if P is in T. If-u^M there will be a circle C within T such that at some point P of its boundary, say at Pl9 we have u (P^ = M, whilst in the interior of the circle u < M . * E. Picard, T mitt tf Analyse, t. ir, 2nd ed., p. 29 (Paris, 1905). t S. Bernstein, Math. Ann. Bd. ux, S. 69 (1904). J L. Lichtenstein, Palermo Rend. t. xxxm, p. 211 (1912); Math. Zeitschr. vol. xx, p. 205 (1924). § E. Hopf, Berlin. Sitzungsber. S. 147 (1927). 136 Applications of the Integral Theorems of Gauss and Stokes Let K be a circular realm of radius R whose circular boundary touches C internally at P, then with the exception of the point Pl , we have every- where in K the inequality u < M. Next let a circle K1 of radius /^ < R be drawn so as to lie entirely within T. The boundary of K± then consists of an arc St (the end points included) which belongs to K and an arc $0 which does not belong to K. On St we have the inequality u < M — e, where e is a suitable small quantity, while on S0 we have u ^ M. ...... (A) We now 'choose the centre of K as origin and consider the function h(P) == e-'1'2- e-i;-", where r2 = a:2 -f ?/2 and a > 0. If xt = x, x2 = y and £ (M) - .4w^ -f 2£?^y -f CM,, -f Dux -f J^, a simple calculation gives e^L (h) = 4a2 (Ax2 -f 2&oy -f Cy2) - 2a (A + C -f Dz -f %). Since the equation is of elliptic type we have in the interior and on the boundary of K . 0 rt „ ~ 9 ^ 7 ^ J J.r2 -f 2Bxy -f <7?/2 > & > 0, where k is a suitable constant. By choosing a sufficiently large value of a we can make r , 7 , . L (//) ^> 0 in Kl and so L (u 4- 8A) > 0 if 8 > 0. We have, moreover, h (P) < 0 when P is on $0, & (Pj) = 0. ..... (B) We now put v (P) = u (P) -f- 8 . h (P), 8 > 0, where 8 is also chosen so small that, in view of (A), we have v < M on Sz. On account of (A) and (B) we have further v < M on SQ. Hence v< M on the whole of the boundary of K1 and at the centre we have v = M . Thus v should have a maximum value at some point in the interior of Kl . This, however, may be shown to be incompatible with the inequality L (v) > 0, for at a place where v is a maximum we have by the usual rule of the differential calculus 0 for arbitrary real values of A and ^. Now, by hypothesis, therefore by the theorem of Paraf and Fejer (§ 1-35) Avxx + 2Bvxy + Bvyv < 0. But the expression on the left-hand side is precisely L (v) since vx == vv — 0, and so we have L (v) < 0 which is incompatible with L (v) > 0. The case in which L (u) < 0, u (P) > u (P0) can be treated in a similar way. Maxima and Minima of Solutions 137 In particular, if L (u) = 0 in T, where u is not constant, neither of the inequalities u (P) < u (P0), u (P) > u (P0) can hold throughout T when P i& an internal point. This means that u (P) cannot have a maximum or minimum value in the interior of a region T within which it is regular. This theorem has been extended by Hopf to the case in which the fr notions A, B, C, D, E are not continuous throughout T but are bounded functions such that an inequality of type AX2 + 2BXfM + C>2 > N (A2 + n*) > 0 holds, with a suitable value of the constant N, for all real values of A and H and for all points P in T. The work of Picard has also been generalised by Moutard* and Fejerf. The latter gives the theorem the following form : Let n, ... n n S aik(x1)x2) ...xn)(u)tk+ I>br(xl9x29 ... xn) (u)r + c (xl9 ...xn)u= 0, i,...i i atk (Xl> %2> ••• Xn) — aJft (X\> ^2? ••• %n) be a homogeneous linear partial differential equation of the second order with real independent variables xl9x2, ... xn and a real unknown function u (xl9 x2) ... xn). The coefficients ^ik \%lj *^2> •*• ^n/9 ®r \%l j *^2 > ••• ^n/9 ^ (^1 > ^2 > ••• *^n) are all real functions which can be expanded in convergent power series of yP6S c (xl9 *2, ... xn) - c + Cj^ + ... cnxn + cnx^ -f ... , 6r (a?!, #2, ... xn) = 6r -f &rl#i + ... ftrn^n -f brllxf + .... «fjfc (^1, ^ -• *n) = «tA: + GW^l + ..-, for \Xi\<*i, I ^2 I < ^2» — I %n | < ««, where zl9 z2, ... 0n are suitable constants. Then, if n, n 2 atkytyk > 0 1,1 for all real values of yl9 y2, ... yny that is, if the quadratic form is non- negative, and if c < 0, the differential equation has no solution which is regular at the origin and has there either a negative minimum or a positive maximum. If, however, the quadratic form is not negative, that is, if 71, 71 2 at*iM* < 0 for some set of values rjl9 ^2, ... rjk9 there is always a solution regular at the origin which, if c < 0, has either a negative minimum or a positive maximum. Thus when c< 0 the requirement that the quadratic form * Th. Moutard, Journ. de Vtfcole Poll/technique, t. LXIV, p. 55 (1894); see also A. Paraf, Annates de Toulouse, t. vi, H, p. 1 (1892). t Loc. cit. 138 Applications of the Integral Theorems of Gauss and Stokes should not be of the non-negative type is a necessary and sufficient con- dition for the existence of a negative minimum or positive maximum at the origin for some regular solution of the differential equation. § 2-31. Green's theorem for Laplace's equation. Let us now write in equation (A) of § 2-11 the equation then takes the form dU dV dU dV dU dV\ rrdv\jv trrwrj t ( d \ U v~ }dS = UV2Vdr -Mis- -x-- + ~- 3- 4- -5- -~- on/ J J \ox dx dy dy dz dz J dV where the symbol ~-- is used for the normal component of VF, dV dV dV dV Interchanging U and V we have likewise ((vdu\jv (VWTJ _L [fiUdVdUdV dUdV I K 5— ) a£ = y\2Udr + - -=-- + - - -^- - - -^- — - J \ onj ) ) \ox dx dy dy dz dz Subtracting we obtain Green's theorem, In this equation the functions U and V are supposed to be continuous (D, 2) within the region over which the volume integrals are taken. This supposition is really too restrictive but it will be replaced later by one which is not quite so restrictive. If the functions U and V are solutions of the same differential equation and one which has the same characteristic as Laplace's equation, an interesting result is obtained. In particular, if k*U - 0, where k is either a constant or a function of x, y and z, the volume integral vanishes and we have the relation In the special case when the surface S is a sphere and U = fm(r)Tm(**g\ V = fn(r)Yn(HZ}, where Ym and Yn are functions which are continuous over the sphere and fm (r), fn (r) are f unctions such that fm (r) fn' (r) ^ fm' (r) fn (r) on the sphere we obtain the important integral relation which implies that the functions Ym form an orthogonal system. Green's Theorem for Laplace's Equation 139 An appropriate set of such functions will be constructed in § 6-34. When k is a constant the equations (B) may be derived from the wave- equations D2^ = 0, D2^ = 0 by supposing that u and v have the forms u = U sin (kct -ha), v = V sin (kct -f /?) respectively. When k is real these wave -functions are periodic. When k = 0, U and V are solutions of Laplace's equation. The equation may also be derived from the equation of the conduction of heat, by supposing that this possesses a solution of type v = e~*m V(x, y, z). Green's theorem is particularly useful for proofs of the uniqueness of the solution of a boundary problem for one of our differential equations. Suppose, for instance, that we wish to find a solution of Laplace's equation which is continuous (Z), 2) within the region bounded by the surface S and which takes an assigned value F (x, y, z) on the boundary of S. If there are two such solutions U and F the difference W = U — V will be a solution of Laplace's equation which is zero on the boundary and continuous (D, 2) within the region bounded by 8. Green's theorem now gives n fw zw ,a (T/swY /3WV fiw\*\ i 0 == \w --- dS = ( -3 - - ) + K- ) + hr ) dr> J dn }[\dx J \dyj \ dz ) J and this equation implies that dW_ y" ' dz ~Uj W is consequently constant and therefore equal to its boundary value zero. Hence U = V an|J the solution of the problem is unique. A similar con- dW dW elusion may be drawn if the boundary condition is ^~ =0 or -~ — h h W = 0, where h is positive. If the equation is V2F -h AF = 0 instead of Laplace's equation the foregoing argument still holds when A is negative, for we have the additional term — A I W2dr on the right-hand side. The argument breaks down, however, when A is positive. In the case of the equation of heat conduction (C) there are some similar theorems relating to the uniqueness of solutions. If possible, let 9?^ _ there be two independent solutions vly v2 of the equation ^ = *:V2t> and the supplementary conditions v=f(x, y, z) for t = 0 for points within S, v ~ <f> (x, y, z, t) on S (t > 0), v continuous (J5, 2) within region bounded by S. 140 Applications of the Integral Theorems of Gauss and Stokes Let V = v1 - v2, then F = 0 f or t = 0 within S, V = 0 on S. Putting 27 - [ F2dr, we have ^ = f F 3/dr = /c I V (V2F) dr c# J 9£ ' -I T/^F 70 [[fiV\* /3F\2 /3 7 F .- AS - * U ) + br ) + ^ \dr' dn ] [\Sx/ \dyj \ dz Since F = 0 on S the first integral vanishes, and so dl /T/^V /3F\2 /SV\2] 7 /,^AX ~. - - AC ---- ) -I U, - ) + ~ rfr * > 0). c# J [_\ do; / \G7// \ c/2 / J But 7 = 0 for t = 0, therefore 7 < 0, but on the other hand the integral for 7 indicates that 7 > 0, consequently we must have 7=0, F = 0. These theorems prove the uniqueness of solutions of certain boundary problems but they do not show that such solutions exist. Many existence theorems have been established by the methods of advanced analysis and the literature on this subject is now very extensive. § 2-32. Green's functions. The solution of a problem in which a solution of Laplace's equation or a periodic wave-function is to be determined from a knowledge of its behaviour at certain boundaries can be made to depend on that of another problem — the determination of the appropriate Green's function*. Let Q(x, y, z-9xl9 yl,z1) be a solution of V2G + k2G = 0 with the following properties: It is finite and continuous (Dy 2) with respect to either x, y, z or xl , yl , z1 in a region bounded by a surface S, except in the neighbourhood of the point (x19 yl9 zj where it is infinite like T?"1 cos kR as 7? -> 0, R being the distance between the points (x, y,&) and (xl9 yl^ z^). At the surface S9 some boundary condition such as (1) G — 0, (2) dG/dn = 0, or (3) dG/dn -f- hG = 0 is satisfied, h being a positive constant. Adopting the notation of Plemeljt and KneserJ, we shall denote the values of a function <j> (f, 77, £) at the points (x, y, z)9 (xl9 yly zt) respectively by the symbols </> (0) and <£(!). When a function like the Green's function depends upon the co- ordinates of both points it will be denoted by a symbol such as G (0, 1). The importance of the Green's function depends chiefly upon the following theorem : Let U be a solution of V2C7 + k*U + 47T/ (x, y, z) - 0, (A) * G. Green, Math. Papers, p. 31. t Monatshefte far Math. u. Phys. Bd. xv, S. 337 (1904). J A. Kneser, Die Integralgleichungen und ihre Anwendungen in der mathematischen Physik (Vieweg, Brunswick, 1911). Green's Functions 141 which is finite and continuous (Z>, 2) throughout the interior of a region ® bounded by a surface 8 and let / (x, y, z) be a function which is finite and continuous throughout !3). We shall also allow / (x, y, z) to be finite and continuous throughout parts of the region and zero elsewhere. Applying Green's theorem to the region between a small sphere whose centre is at (x± , yl , zj and the surface S, we have Now V2(? — — k2G and the first integral on the right may be found by a simple extension of the analysis already used in a similar case when G = 1/7?, consequently we have the equation (1) = (O, I)/ (0)dT0 + --GddS0 ....... (B) If U satisfies the same boundary conditions as G on the surface 8 the surface integral vanishes and we have* '0. (C) If, on the other hand, / (x, y, z) = 0 and G = 0 on 8 we have (D) the value of U is thus determined completely and uniquely by its boundary o/nr values. Similarly, if the boundary condition is ~ = 0 on S we have on ?„, (E) and the value of U is determined by the boundary values of dU/dn. Finally, if the boundary condition satisfied by 0 is «- + hG = 0, we have (F) and U is expressed in terms of the boundary values of ^- — f- hU. If g (x2, 2/2, z2; x, y, z) is the Green's function for the same boundary condition as G (0, 1) but for the value a of &, we must also surround the * It has not been proved that whenever the function / is finite and continuous throughout D the formula (C) gives a solution of (A). Petrmi has shown in fact that when/ is merely continuous the second derivatives of the integral may not exist or may not be finite. Ada Math. t. xxxr, p. 127 (1908). It should be remarked that Gauss in 1840 derived Poisson's equation (§2-61) on the supposition that the density function/ is continuous (/>, 1). With this supposition (C) does give a solution of (A). Poisson's equation and the solution of (A) are usually derived now for the case of a function / which satisfies a Holder condition. See Kcllogg's Foundations of Potential Theory, ch. vi. 142 Applications of the Integral Theorems of Gauss and Stokes point (#2, y*-, z*) by a small sphere when we apply Green's theorem with U (x, y, z) = g (2, 0). We then obtain the equation /£2 _ (jl^ ,' g (2, 1) = G (2, 1) - L_J g (2, 0) G (0, 1) dr, ....... (G) This may be regarded as an integral equation for the determination of g (2, 0) when G (0, 1) is known or for the determination of G (0, 1) when g (2, 1) is known. In some cases the Green's function for Laplace's equation (k = 0) can be found and then the integral equation can be used to calculate g (2, 0) or to establish its existence. The Green's function for Laplace's equation, when it exists, is unique, for if G (0, 1), H (0, 1) were two different Green's functions the function V (0) = G (0, 1) - H (0, 1) would be continuous (D, 2) throughout the region bounded by the surface S and satisfy the boundary condition that was assigned, but such a function is known to be zero. For small values of a2 the function g (2, 0) can be obtained by expanding it in the form g (^ Q) = fl (2> Q) + ^ (^ Q) + .......... (R) The first term is the corresponding Green's function for Laplace's equation and is known, the other terms may be obtained successively by substituting the series in the integral equation (with k = 0) and equating coefficients of the different powers of a2. 80 long as the series converges this method gives a unique value of g (2, 0). The value of g (2, 0), if it exists, will certainly not be unique when a2 has a singular or characteristic value for which the "homogeneous integral equation" has a solution </> which is different from zero. In this case 4rr</> (1) = (a2 - i«) JV (0) G (0, 1) rfr0, ...... (I) and the formula (C) indicates that this function <£ (0) = U (x, y, z) is a solution of V2£/-fcr2t/=0, which satisfies the assigned boundary con- ditions and the other conditions imposed on U. The solutions of this type are of great importance in many branches of mathematical physics, particularly in the theory of vibrations, and have been discussed by many writers. The characteristic values of a2 are called Eigenwerte by the Germans and the corresponding functions </> Eigenfunktionen. These terms are now being used by American writers, but it seems worth while to shorten them and use eit in place of Eigenwert and eif in place of Eigen- funktion. The same terms may be used also in connection with the homogeneous integral equation (I). In discussing this equation it is convenient, however, to put k = 0, so that G becomes the Green's function Symmetry of a Green's Function 143 for Laplace's equation and the assigned boundary conditions. Denoting this function by the symbol 4-rrK (0, 1) we have the integral equation (0) K (0, 1) dr0 for the determination of the solution of V2<f> + cr2<f> = 0 and the assigned boundary conditions, that is, for the determination of the eifs and eits. The function K (0, 1) is called the kernel of the integral equation; it has the important property of symmetry expressed by the equation K (0, 1) = K (1, 0). This may be seen as follows. If we put 4rr/ (0) = (o-2 — k2) g (0, 2) in the formula (B) and proceed as before, Green's theorem gives g (I, 2) = 0 (2, 1) - -^^ /</ (0, 2) O (0, 1) dr0. Putting o- = k and comparing this equation with the previous one we obtain the desired relation. When k ^ 0 the relation gives 0(1, 2) = 0(2,1). When the boundary condition is ~- = 0 this result is equivalent to one (j'Yl given by Helmholtz in the theory of sound. If \fj (0) is an eif corresponding to an eit v2 which is different from a2 we have I (1) = v2 (0) K (0, 1) dr0, and, if the order of integration can be changed, K(0, = „« <£ (0) I (0) dr. = v« (1) * (1) drj. Hence the eifs <£ and «/r satisfy the orthogonal relation This result may be used to prove that the eits a2 are all real. If, indeed, a2 were a complex quantity a + if! the corresponding eif <f> (0) would also be a complex quantity x (0) + i<*> (0), and since K is real the function 0 (0) = x (0) — *<*> (0) would be an eif corresponding to the eit v2 = a — i/J, and the orthogonal relation 0 = jt (0) 0 (0) <*TO = |{[X (O)]2 + [co (0)]*} dr. would be satisfied. This, however, is impossible because the integrand is 144 Applications of the Integral Theorems of Gauss and Stokes either zero for all values of the variables or positive for some, but it is zero only when x (0) - «, (0) - 0, * (0) - 0. To prove that the eits are all positive we make use of the equation * j U'dr = - f U Vffrfr- f \(f-)2 4- (^)2 + (f )'] dr. J J J l\ox/ \dy / \dz / J The Green's function is usually found in practice by finding the eifs and eits directly from the differential equation and then writing down a suitable expansion for G in terms of these eifs. The question of convergence is, however, a difficult one which needs careful study. The method has been used with considerable success by Heine in his Kugelfunktionen, by Hilbert and his co-workers, by Sommerfeld, Kneser and Macdonald, § 2-33. Partial difference equations. The partial difference equations analogous to the partial differential equations satisfied by conjugate functions are 7/ _ ., ,,, _ -, ux — "Hi My — ~ vx> and these lead to the equations of § 1-62 uxx -f- uvv = 0, Vxz 4- Vyy = 0, which are analogous to Laplace's equation. These difference equations have been used in recent years to find approximate solutions of Laplace's equation when certain boundary conditions are prescribed* and also to establish the existence of a solution corresponding to prescribed boundary conditions. Let us consider, for instance, a square whose sides are x — ± 2 h, y — -{_ 2h and let us introduce the abbreviations a = h, b = 2h, a = — h, j8 = — 2h, u (x, y) = (xy), u(b,y)=(y), u(p,y) = (y); u(x,b)=[x], u (x, p) = [x], then we have eight non-homogeneous equations (fi-A-equations) - (Oa) - («0) + 4 (aa) = (a) -f [a], (Oa) + (aO) - 4 (aa) - (a) -f [a], - («0) - (Oa) -f- 4 (aa) - (a) -f [a] , (Oa) -f (aO) - 4 (aa) - (a) + [a], (aa) + (00) -f (aa) - 4 (aO) = - (0), (act) + (00) + (aa) - 4 (aO) - - (0), (aa) + (00) + (aa) - 4 (Oa) = - [0], (aa) + (00) -f (aa) - 4 (Oa) = - [0], and one homogeneous equation (A-equation) (Oa) 4- (aO) 4- (Oa) 4- (aO) - 4 (00). The first step in the solution is to eliminate the quantities (aa), (aa), (aa), (aa) which do not occur in the ^-equation. This gives the equations 4 (00) 4- (Oa) 4- (Oa) - 14 (aO) 4- (a) 4- (a) 4- [a] 4- [a] 4- 4 (6) - 0, 4 (00) 4- (Oa) 4- (Oa) - 14 (aO) 4- (a) 4- (a) 4- [a] 4- [a] 4- 4 (0) = 0, 4 (00) 4- (aO) 4- (aO) - 14 (Oa) 4- (a) 4- [a] 4- (a) 4- [a] + 4 [0] = 0, 4 (00) 4- (aO) 4- (aO) - 14 (Oa) 4- (a) 4- [a] 4- (a) + [a] + 4 [0] = 0. , * L. F. Richardson, Phil. Trans. A, vol. ccx, p. 307 (1911); Math. Gazette (July, 1925). Partial Difference Equations 145 Adding these equations we have - 16 (00) + 12 (aO) -f 12 (Oa) -f 12 (aO) -f 12 (Oa) = 2 (a) + 2 (a) + 2 (a) -f 2 (a) -f 2 [a] -f 2 [a] -f 2 [a] 4- 2 [a] + 4(0) + 4(0) + 4[0]+4[6]. Combining this with the homogeneous equation we see that the quantity on the right-hand side of the last equation is equal to -f 32 (00) and so the quantity (00) is obtained uniquely. Similarly, if the sides of the square are x = ± 3h, y = ± 3h there are 16 7i-A-equations and 9 A-equations which may be solved by first eliminating the quantities which do not occur in the /^-equations. We have then to solve 9 linear equations in order to obtain the remaining quantities, but these 9 equations may be treated in exactly the same way as the previous set of 9 linear equations, quantities being eliminated which do not occur in the central equation. In this way a value is finally found for (00). A similar method may be used for a more general type of square net- work or lattice. Let the four points (x -f- h, y), (x — h, y), (x, y ,+ /*)» (x, y — h) be called the neighbours of the point (x, y) and let the lattice L consist of interior points P, each of which has four neighbours belonging to the lattice, and boundary points Q, each of which has at least one neighbour belonging to the lattice and at least one neighbour which does not belong to the lattice. A chain of lattice points Alt A2, ... An+1 is said to be connected when ^4S+1 is one of the neighbours of As for each value of ^ in the series 1, 2, ... n. A lattice L is said to be connected when any two of its points belong to a connected chain of lattice points, whether the two points are interior points or boundary points. The lattice has a simple boundary when any two boundary points belong to a chain for which no two consecutive points are internal points and no internal point P is consecutive to two boundary points having the same x or the same y as P. The solubility of the set of linear equations represented by the equation ux-x -f uyy = 0 (A) for such a lattice may be inferred from the fact that this set of linear equations is associated with a certain quadratic form h2 2 (ux* f uj)9 where the summation extends over all the lattice points, and a difference quotient associated with a boundary point is regarded as zero when a point not belonging to the lattice would be needed for its definition. This sum- mation can, by the so-called Green's formula, be expressed in the form - WJ^u(ux2 + uvy) - h^uR(u), (B) P Q where the boundary expression R (u) associated with a boundary point UQ is defined by the equation hR (UQ) — u± -f u2 -h ... us — &UQ, 146 Applications of the Integral Theorems of Gauss and Stokes where u^u^, ... UB are the 8 neighbouring points of UQ (s < 3). Since uxx -f uv? = 0 the quadratic form can be expressed in terms of boundary values. If there were two solutions of the partial difference equation with the same boundary values, the foregoing identity could be applied to their difference u — vy and since the boundary values of u — v are all zero the identity would give the relation 2 [(*„ - *.)• + (u, - vyy] = o, which implies that ux — vx = 0, uy — vy = 0 for all points (x, y) of the lattice ; consequently, since u — v is zero on the boundary it must be zero throughout the lattice. There is another identity 0 = A2 2 (vuxx -f vUyj — uvxx — uVyy) + h 2 [vR (u) — uR (v)] P Q which, when applied to the case in which uxx -f uyy = 0 and the boundary^ value of v is zero, gives 2 [(ux + vx)* + (uy + vy)*] = - 2 (u + v) (vxx + vvy) - h~l S (u) [R (v) + R (u)] P Q = - A-1 S [vR (u) - uR (v)] + 2 (vx* + vv2 + ux* + uy2) Q + A-1 2 uR (u) - h-1 2 [uR (u) + uR (v)] Q Q = S (v.» + vy2 + ux* + uy*) > S (uj + uy*), the transformations being made with the aid of Green's formula (B). This equation shows that the solution of uxx -p uyy = 0 and the pre- scribed boundary condition gives the least possible value to the quadratic form. The system of linear equations uxx -f uyy = 0 can, indeed, be ob- tained by writing down the conditions that the quadratic form should be a minimum when the boundary values of u are assigned. With a change of notation the quadratic form may be written in the form N N N 2 2 cmnumun - 2 2 anun + 6, m»l n— 1 n— 1 where the quadratic form is never negative. The corresponding set of linear equations H -f ciaw2 -f- ... CINUN - a1? has a determinant | cmn | which is not zero and so can be solved. For the sake of illustration we consider a lattice in which the internal lattice points are represented in the diagram by the corresponding values Associated Quadratic Form 147 of the variable u and the boundary lattice points by corresponding values denoted by t/s. The quadratic form is in this case K-*g2+K-^i)2+K-^ to - *>e)2 4- K - *4>)2 + («o - ^io)2 + K - ^)2 + to - ^i)2 + to - t>9)2, (t>3 - ^2)2 + K - ^8)2 4- (K, ~ ^4)2 + (*4 ~ "5)2> and it is easy to see that the equations obtained by differentiating with respect to u± , u2 respectively are 4^j = u0 + u2 + ^ -f v2J 4^2 = ^ + 1*3+^5 + V3> and are of the required type. The quadratic form is, moreover, equal to the sum of the quantities t^-Wa-VM-tJoJ+M^ - <NO - w2 - u4 - v9) 4- w4 (4w4 -w3 -i*5 - v1 - vB)+u6 (4w6 -^2 -^4-^-^4), ^o («o - wo) 4- Vi («i ~ ^) + v2 (^2 - ^i) 4- % to ~ ^2) 4- ^4 (^4 - ««») -f v6 to - ^s) 4- v? (v? - %) 4- vs (vg - u4) + v9 (w9 - Ws) + v10 (VM - ^0)- § 2-34. TAe limiting process^. We assume that 6 is a simply connected region in the ^-plane with a boundary F formed of a finite number of arcs with continuously turning tangents. If v is an integrable function defined within G we shall use the symbol 0 {v} to denote the integral of v over the area G and a similar notation will be used for integrals of v over portions of G which are denoted by capital letters. Let Gh be the lattice region associated with the mesh-width h and the region G, and let the symbol Gh [v] be used to denote the sum of the values of v over the lattice points of G. Also let the symbol I\ (v) be used for the sum of the values of v over the boundary points which form the boundary I\ of Gh. This notation will be used also for a portion of Gh denoted by a capital letter and for the lattice region Qh* belonging to a partial region Q* of G. Now let / (x, y) be a given function which is continuous (D, 2) in a region enclosing G and let u (#, y) be the solution of (A) which takes the same value as / (#, y) at the boundary points of Gh. We shall prove that as h -> 0 the function uh (x, y) converges towards a function u (xy y) which f R. Courant, K. Friedrichs and H. Lewy, Math. Ann. vol. c, p. 32 (1928). See also J. le Roux, Journ. de Math. (6), vol. x, p. 189 (1914); R» G. D. Richardson, Trans. Amer. Math. Soc. vol. xvm, p. 489 (1917); H. B. Phillips and N. Wiener, Journ. Math, and Phys. Mass. Inst. Tech. vol. n, p. 105 (1923). 10-2 148 Applications of the Integral Theorems of Gauss and Stokes satisfies the partial differential equation V2u = 0 and takes the same value as / (#, y) at each of the points of F. We shall further show that for any region lying entirely within G the difference quotients of uh of arbitrary order tend uniformly towards the corresponding partial derivatives of u (x, y). In the convergence proof it is convenient to replace the boundary condition u — f on F by the weaker requirement that <$>{(^~/)2}-*0 asr->0, where Sr is that strip of G whose points are at a distance from F smaller than r. The convergence proof depends on the fact that for any partial region G* lying entirely within G, the function uh (x, y) and each of its difference quotients remains bounded and uniformly continuous as h -> 0, where uniform continuity is given the following meaning : There is for any of these functions wh (x, y) a quantity 8 (e), depending only on the region and not on h, such that if wh(p) denote the value of the function at the point P we have the inequality I *VP) - *^(PI) | < * whenever the two lattice points P and Pl of the lattice region Gh lie in the same partial region and are separated by a distance less than 8 (e). As soon as the foregoing type of uniform continuity has been established we can in a well-known manner f select from our functions uh a partial sequence of functions which tend uniformly in any partial region 6?* towards a limit function u (x, y} while the difference quotients of uh tend uniformly towards that of u (x, y} differential coefficients. The limit func- tion then possesses derivatives of order n in any partial region G* of G and satisfies V*u = 0 in this region. If we can also show that u satisfies the boundary condition we can regard it as the solution of our boundary problem for the region. G. Since this solution is uniquely determined, it appears then that not only a partial sequence of the functions uh but this sequence of functions itself possesses the desired convergence property as A-+0. The uniform continuity of our quantities may be established by proving the following lemmas : (1) As h -* 0 the sums h*Qh [u*] and h*Gh [ux2 + uyz] remain bounded. (2) If w = wh satisfies the difference equation (A) at a lattice point of Gh and if, as h -> 0 the sum h2G A* [w2] , extended over a lattice region Gh* associated with a partial region G* of G, remains bounded, then for any fixed partial region 6?** lying entirely within 6?* the sum f See for instance, Kellogg's Foundations of Potential Theory, p. 265. The theorem to be used is known as Ascoli's theorem; it is discussed in § 4-45. Inequalities 149 over the lattice region 6?A** associated with 6**, likewise remains bounded as Ti-> 0. When this is combined with (1) it follows that, because all the difference quotients w of the function uh also satisfy the difference equation (A), each of the sums h2Gh* [w2] is bounded. (3) From the boundedness of these sums it follows that the difference quotients themselves are bounded and uniformly continuous as h -> 0. The proof of (1) follows from the fact that the functional values uh are themselves bounded. For the greatest (or least) value of the function is assumed on the boundary *f and so tends towards a prescribed finite value. The boundedness of the sum h2Gh [ux2 -f uy2] is an immediate consequence of the minimum property of our lattice function which gives in particular h*Gh [ux2 + uy2] < WQK (fx2 + A2], but as h-> 0 the sum on the right tends to G \( J ) -f ( ~ ) ,v, which, by [\dx/ \vy>) hypothesis, exists. To prove (2) we consider the sum h2Ql [ivx2 -f w^ + wy2 -f- w^2], where the summation extends over all the interior points of a square Qt. Now Green's formula gives h2Qi K2 + ™** + Wy2 + ?V] = 2 (w2) - S (w2), 1 0 where £j and 20 are respectively the boundary of Qv and the square boundary of the lattice points lying within Sj . We now consider a series of concentric squares Q0 , Ql , ... Q v with the boundaries S0, S19 ... 2^v- Applying our formula to each of these squares and observing that we have always 2h2Q0 [wx2 + wy2] < Ji2Qk [w,2 -f w22 + wy2 + wg*] (k > 1), we obtain 2h2Q0 [wx2 + wy2] < 2 (w2) - 2 (iv2) (0 < k < n). A* f 1 k Adding n inequalities of this type we obtain 2nh2Q0 [wx2 -{- wy2] < S (w2) - S (w2) < S (w2). n 0 n Summing this inequality from n = 1 to n = N we get N2h2Q0 [wx2 -f ?V] < QN [w2]. Diminishing the mesh-width h we can make the squares Q0 and QN converge towards two fixed squares lying within 0 and having corre- sponding sides separated by a distance a. In this process Nh -> a and we have independently of the mesh-width />2 h2Q0[wx2^wv2]<a QN[w2]. Uf t On account of equation (A) the value of uh at an internal point is the mean of the values at the four neighbouring points and so cannot be greater than all of them, consequently the greatest value of M* cannot occur at an internal point. 150 Applications of the Integral Theorems of Gauss and Stokes With a sufficiently small value of h this inequality holds with another constant a for any two partial regions of G, one of which lies entirely within the other. Hence the surmise in (2) is proved. To prove that uh and all its partial difference quotients w remain bounded and uniformly continuous as h -> 0 we consider a rectangle R with corners P0, QQJ P, Q and with sides P0QQ, PQ which are x-linesf of length a. Denoting these lines by the symbols XQ , X respectively we start from the formula „ p 7 ir / v , 79r» r -, K^O - wp* = hX (wx) + h*R [wxy] and the inequality | WQQ _ WPQ | < hX (| wx |) + h*R [| wxy |] ...... (C) which is a consequence of it. We now let X vary continuously between an initial position Xl at a distance 6 from XQ and a final position X2 at a distance 26 from X0 and sum the (b/h) -f 1 inequalities (C) associated with X'a which pass through lattice points. We thus obtain the inequality where the summations on the right are extended over the whole rectangle P0Q0P2Q2. By Schwarz's inequality it then follows that | wp» - u#> | < (2a/6)* (A2#2 [wx2])* -f (2a6)i (&2#2 [^xv2])*. Since, by hypothesis, the sums which occur heVe multiplied by A2 remain bounded, it follows that as a -> 0 the difference | wp<> — wA | -^ 0 independently of the mesh- width, since for each partial region G* of 6? the quantity b can be held fixed. Consequently, the uniform continuity of w = wh is proved for the ^-direction. Similarly, it holds for the t/-direction and so also for any partial region (7* of G. The boundedness of the function wh in G* finally follows from its uniform continuity and the boundedness of h*G*[wh*]. By this proof we establish the existence of a partial sequence of functions uh which converge towards a limit function u (x, y) and are, indeed, continuous, together with all their difference quotients, in the sense already explained, for each inner partial region of G. This limit function u (x, y) is thus continuous (Z>, n) throughout (?, where n is arbitrary, tod it satisfies the potential equation In order to prove that the solution fulfils the boundary condition formulated above we shall first of all establish the inequality h2Sr,h [v2] < ArWSfth [vx2 -f V] + Brhrh (v2), ...... (D) where Sf)h is that part of the lattice region Gh which lies within the t This term is used here to denote lines parallel to the axis of x. Properties of the Limit Function 151 boundary strip Sr, which is bounded by F and another curve Fr. The constants A, B depend only on the region and not on the function v or the mesh-width h. To do this we divide the boundary F of G into a finite number of pieces for which the angle of the tangent with either the x or t/-axis is greater than 30°. Let y, for instance, be a piece of F which is sufficiently steep (in the above sense) relative to the #-axis. The x-lines through the end- points of the piece y cut out on Fr a piece yr and together with y and yr enclose a piece sr of the boundary strip Sr . We use the symbol sft h to denote the portion of Gh contained in sr and denote the associated portion of the boundary FA by yh. We now imagine an o?-line to be drawn through a lattice point Ph of Sr th. Let it meet the boundary yh in a point Ph. The portion of this #-line Xh which lies in sr th we call pr th. Its length is certainly smaller than cr, where the constant c depends only on the smallest angle of in- clination of a tangent of y to the #-axis. __ Now between the values of v at Ph and Ph we have the relation vph ^ vph ± hxh (vx). Squaring both sides and applying Schwarz's inequality, we obtain (vf*)* < 2 (iA)2 + 2crhpTfh (vx2). Summing with respect to Ph in the ^-direction, we get Summing again in the ^/-direction we obtain the relation hsr,h [v*] < 2crTh (v?*)* + 2c*r*Sr,h [v,*]. Writing down the inequalities associated with the other portions of F and adding all the inequalities together we obtain the desired inequality (D). By similar reasoning we can also establish the inequality h*Qh 0*] < Clhrh (v*) -f c2h*Gh (V + V] in which the constants cls c2 depend only on the region G and not on the mesh division. We now put vh = uh — fh so that vh = 0 on FA . Then, since h2Gh [vx2 -f vy2] remains bounded as h -> 0, we obtain from (D) (h*/r)Srtk[v*]<Kr, ...... (E) where K is a constant which does not depend on the function v or the mesh- width. Extending the sum on the left to the difference S^h — Spfh of two boundary strips, the inequality (E) still holds with the same constant K and we can pass to the limit h -* 0. From the inequality (D) we then get (l/r)S[v*]<Kr, 152 Applications of the Integral Theorems of Gauss and Stokes where 8 — Sr — Sfl and v — u — /. Now letting /> -> 0, we obtain the inequality ^ ^ ^ ^ (1/jp) ^ [(u _ /)2] ^^ ^ which signifies that the limit function satisfies the prescribed boundary condition. § 2-41. The derivation of physical equations from a variational principle. \ concise expression may be given to the principles from which an equation or set of equations is derived by using the ideas of the " Calculus of Variations*." This expression is useful for several purposes. In the first place a few methods are now available for the direct solution of problems in the "Calculus of Variations" and these can sometimes be used with advantage when the differential equations are hard to solve. Secondly, when an integral's first variation furnishes the desired physical equations the expression under the integral sign may be used with advantage to obtain a transformation of the physical equations to a new set of co- ordinates, for the transformation of the integral is generally much easier than the transformation of the differential equations and the transformed equations can generally be derived from the transformed integral by the methods of the "Calculus of Variations," that is, by the Eulerian rule. To illustrate the method we consider the variation of the integral r i f ff [f3v\2 isv\2 /2F /== 6 a 'I + "a -) + br 2JJJ [\dxj \dy/ \dz when the dependent variable V is alone varied and its variation is chosen so that it vanishes on the boundary of the region of integration. We have Now by a fundamental property of the signs of variation and differentia- tion a I/ a ^ 0 V 0 ~J7 8-x— — Q- (SF), etc. dx 8x v h Hence 87 - f I f \~ j^ (8V) + ...1 dxdydz V~dS- \8V.V2Vdxdydz. dn J J J y The surface integral vanishes because 8F = 0 on the boundary, conse- quently the first variation 87 vanishes altogether if F satisfies everywhere the differential equation .-.^^ ^ V2!/ = 0. The condition 8 F = 0 on the boundary means that as far as the possible variations of F are concerned F is specified on the boundary. It is easily * The reader may obtain a clear grasp of the fundamental ideas from the monograph of G. A. Bliss, "Calculus of Variations," The Carus Mathematical Monographs (1925). Variational Principles 153 seen that a function V with the specified boundary values gives a smaller value of / when it is a solution of V2 V = 0, regular within the region, than if it is any other regular function having the assigned values on the boundary. In the foregoing analysis it is tacitly assumed that V2 V exists and is such that the transformation from the volume integral to the surface integral is valid. If V is assumed to be continuous (Z>, 2) there is no difficulty but, as Du Bois-Reymond pointed out*, it is not evident that a function V which makes 87 = 0 does have second derivatives. This difficulty, which has been emphasised by Hadamardf and LichtensteinJ, has been partly overcome by the work of Haar §. There are in fact some sufficient conditions which indicate when the derivation of the differential equation of a varia- tion problem is permissible. For the corresponding variation problem in one dimension there is a very simple lemma which leads immediately to the desired result. The variation problem is 8 where xl and x2 are constants and 8 V is supposed to be zero for x = x\ and for x — x2 . dV Writing - = M , 8 V = U we have to show that if -« for all admissible functions U which satisfy the conditions U(x,)= U(x2) = 0 ...... (B) then M is a constant (Du Bois-Reymond's Lemma). To prove the lemma we consider the particular function U(x) = (x2 - x) \XM (f) dg - (x - x,) PMtf) dg, J Xt JX which satisfies (B) and gives at any point x where M (x) is continuous - c], say. * P. du Bois-Reymond, Math. Ann. vol. xv, pp. 283, 564 (1879). f J. Hadamard, Comptes Rendus, vol. CXLIV, p. 1092 (1907). J L. Lichtenstein, Math Ann. vol. LXEX, p. 514 (1910). § A. Haar, Journ. fur Math. Bd. CXLIX, S. 1 (1919); Szeged Acta, t. in, p. 224 (1927). Haar shows that in the case of a two-dimensional variation problem the equation 87 = 0 leads to a pair of simultaneous equations of the first order in which there is an auxiliary function W whose elimination would lead to the Eulerian differential equation if the necessary differentiations could be performed. Many inferences may, however, be derived directly from the simultaneous equations without an appeal to the Eulerian equation. 154 Applications of the Integral Theorems of Gauss and Stokes We shall now assume that M(x) is continuous bit by bit (piece wise con- tinuous) so that this equation holds in the interval (xl , x2) except possibly at a finite number of points. The functions M (x), -j- are then undoubtedly (tx integrable over the range (xl9 x2) and, on account of the end conditions (B) satisfied by U(x), we may write (A) in the form With the value adopted for U this equation becomes cxt [M (x) — c]2 dx = 0, dM d2V and implies that M (x) — c, hence -7- = 0 and -v-z = 0. a# dx2 An extension of this analysis to the three-dimensional case is difficult. To avoid this difficulty it is customary* to limit the variation problem and to consider only functions that are continuous (D, 2) throughout the region of integration. The function V and the comparison function V + U are supposed to belong to the field of functions with the foregoing property. The problem is to find, if possible, a function V of the field such that 81 is zero whenever U belongs to the field and is zero on the boundary of the region of integration. Even when the problem is presented in this restricted form a lemma is needed to show that V necessarily satisfies the differential equation. We have, in fact, to show that if U.V2V dxdydz^ 0, for all admissible functions U, then V2 V = 0. The nature of the proof may be made clear by considering the one- dimensional case. We then have the equations <f) (x) U (x) dx = 0, and the conditions : U (x) is continuous (Z>, 2), <f>(x) is continuous in (xl9 x2). Since U (x) is otherwise arbitrary we may choose the particular function U (x) = (x — a)4 (b — #)4 xl < a < x < 6 < x2 = 0 otherwise. If <f>(x) were not zero throughout the interval (xlf x2) it would have a definite sign (positive, say) in some interval (a, 6) contained within (xl9 #2), * See, for instance, Hilbert-Courant, Methoden der Mathematischen Physik, vol. I, p. 165. Fundamental Lemmas 155 but this is impossible because with the above form of U(x) the integral <f> (x) U(x)dx is positive. J X\ i To extend this lemma to the three-dimensional problem it is sufficient to consider a function U(x, y, z) which has a form such as within a small cube with (alf a2, a3), (blt 62> ^3) as ends of a diagonal, the value of U outside the cube being zero. In this way it can be shown that a field function V for which 81 — 0 is necessarily a solution of V2 V = 0. The foregoing analysis does not prove, however, that such a function exists. Similar analysis may be used to derive the equation V2<f> + k*(f> = 0 from a variational principle in which 8 If \Ldxdydz = 0, When the potential <f> is of the form - cos kr the volume integral is finite although the integral k2<f>2 is not*. EXAMPLES i. if /ass f/T(li)8" the equation 87 = 0 may be satisfied by making V — f(x + y) + g(x — y) where / and g have first derivatives but not necessarily second derivatives. [Hadamard.] 2. The variation problem 8 F(VX, Vy, x, y)dxdy = 0 leads to the simultaneous equations W-dF W -~™~ x~dVy' v~ BVX' the suffixes x, y denoting differentiations with respect to these variables. [A. Haar.] § 2»42. The general Eulerian rule. To formulate the general rule for finding the equations which express that the first variation of an integral is zero we consider the variation of an integral f f f / = ... L dxl dx2 . . . dxn , where L is a function of certain quantities and their derivatives. For , * See, for instance, the remarks made by J. Lennard- Jones, Proc. London Math. Soc. vol. xx, p. 347 (1922). 156 Applications of the Integral Theorems of Gauss and Stokes brevity we use Suffixes 1, 2, etc. to denote derivatives with respect to #!, x2, etc. If there are m quantities u, v, w, ... which are varied inde- pendently except for certain conditions at the boundary of the region of integration, there are m Eulerian equations which are all of type o = _ - - - - ~ e du S,idx3\duj 2la^it^i dxadxt\dust) at 1 " " n 33 / SL ~ 3 ! r?t .?! £ aavaar.a*; l these are often called the Euler-Lagrange differential equations, but for brevity we shall call them simply the Eulerian equations. If /j , 12 , . . . ln are the direction cosines of the normal to an element of the boundary, the boundary conditions are of types , ' rdL i a / a/A i a» / a/A i ldu. 2!2iaTAe"a^J + 3!2ja*;aa:Ae"'9«rJ "T r-l 0= S a-! There are m boundary conditions of the first type, mn boundary con- ditions of the second type, ^mn(n— 1) boundary conditions of the third type, and so on. In these equations the coefficients $st, €rst are constants which are defined as follows : €8t - 1 S ^ t, = 2 8= t, erst = 1 r ^ s ^ t, = 2 r = s ^ t, = 6 r = s ^ t. The equations (A) are obtained by subjecting the integral to repeated integrations by parts until one part of the integral is an integral over the boundary and the other part is of type ||... l[USu+ V8v+ WSw+ ...]dx, ... dxn. The equations ^ ^ y _ ^ w = Q? are then the Eulerian differential equations*, while the boundary integral is (dS [USu + S UtSut + L UnSurt + ...], and the boundary conditions are f/ = 0, f/(=0 («=1, 2,...), f/rt=0 (r= 1,2, ...;<= 1,2,...). * For general properties of the Eulerian equations see Ex. 2, p. 183, and the remarks at the end of the chapter. The Eulerian Equations 157 Typical integrations by parts are ^ d [su — 1 -ou a (SL\ a r ail j) r, l_/^\] s ^2 /^\ a2 /dL rs 3L] a ra a /3L\i , a °UI i \ ~ ^ \ °u ^ - \ ^ }\ + OU ~ [_ ^Uu\ vxi L dx2\duu/ \ dx^ ~ a£ __ a r\ 3//1 a r a /a//\~] ~ d* /dL' The reason for the introduction of the factors tst is now apparent. When L depends only on a single quantity u and its first derivatives the Eulerian equation is of the second order. The variation problem is then said to be regular when this partial differential equation is of elliptic type. The distinction between regular and irregular variation problems becomes apparent when terms involving the square of 8u are retained and the sign of the sum of these terms is investigated (Legendre's rule). When a variation problem is irregular it is not certain that the boundary conditions suggested by the variation pioblem will be equivalent to those which are indicated by physical considerations. For a physically correct variation problem a direct method of solution is often advantageous. The well-known method of Rayleigh and Ritz is essentially a method of approximation in which the unknown function is approximated by a finite series of functions, each of which satisfies the specified boundary conditions. The coefficients in the series are chosen so as to make 81 = 0 when each coefficient is varied. The problem is thus reduced to an algebraic problem. § 2-431. The transformation of physical equations. In searching for simple solutions of the partial differential equations of physics it is often useful to transform the equations to a new set of co-ordinates and to look for solutions which are simple functions of these co-ordinates. The necessary transformations can be made without difficulty by the rules of tensor analysis and the absolute calculus, but sometimes they may be obtained very conveniently by transforming to the new co-ordinates the integral which occurs in a variational problem from which they are derived. The principle which is used here is that the Eulerian equations which are derived from the transformed integral must be equivalent to the Eulerian equations which were derived from the original integral because each set of equations means the same thing, namely, that the first variation of the integral is zero. A formal proof of the general theorem of the covariance of the Eulerian equations can, of course, be given *, but in this book we shall * L. Koschmieder, Math. Zeits. Bd. xxiv, S. 181, Bd. xxv, S. 74 (1926); Hilbert and Courant, I.e. p. 193. 158 Applications of the Integral Theorems of Gauss and Stokes regard this property of covariance as a postulate. It is well known, of course, that the postulate leads to the Lagrangian equations of motion in the simple case when the integral is of type Ldt, where L = / [qly q2, ... qn\ qlyq2, ... qn] = T - F, the Lagrangian equations being of type dL c 0 = , __. , dq3 dt \SqsJ ' The quantity T here denotes the kinetic energy and V the potential energy. V is a function of the co-ordinates which specify a configuration of the dynamical system, while T is a positive quantity which depends on both the q's and their rates of change, which are* denoted here by g's. In the simple case when n n m IVV/v A A j- = £ 2j Zj drs qr qs , i i n n TT l.VV/» n n 11 where the coefficients ars , crs are constants, the covariance of the equations is easily confirmed by considering a linear transformation of type Ql ^ Qn = Inl9l + ••• Inn9n> in which the coefficients lrs are constants. The advantage of making a transformation is well illustrated by this case, because when the transformation is chosen so that the expressions for T and V take the forms respectively, the Eulerian equations are simply A,Q, + CSQ, = 0, and indicate that there are solutions of type Q. = as cos (n,t + ft), (n*A, = Ct) where as and j8, are arbitrary constants. These co-ordinates Q3 are called the normal co-ordinates for the dynamical problem. Our object now is to see if there are corresponding sets of co-ordinates associated with a partial differential equation. Transformation of Eulerian Equations 159 § 2-432. To transform Laplace's equation to new co-ordinates £, 77, £ such that dx* + dy2 + dz2 = ad^2 + bd^ + cdt* where a, 6, c, /, g, h, are functions of £ , 77, £, we use suffixes a:, t/, z to denote differentiations with respect to x, y, z and suffixes 1, 2, 3 to denote differentiations with respect to f , rj, £. We then have 7 df d^dt, say, . '?» t) «/ say, h - af) = J*F, say. Therefore 2 F3 + 20V3 F, + 2H Vl F2] By Euler's rule 87 = 0 when 9 / 3L\ 9_ / 9L_\ 8 / 3L \ _ a? va v\ ) + ^ V3 F2y + a^ va F3 ) ~ u> The new form of Laplace's equation is thus nv a UV = ^. ^A I •*-* n i -*• oy c/f C/TJ c/4 8 If the original integral is P,a + FV2 + Vz2- XV*]dxdydz, (A) the transformed integral is where U = L — |AF2/J, and so the equation V2F + AF = 0, which is derived from (A) by Euler's rule, transforms into the equation DV + XV IJ = 0 160 Applications of the Integral Theorems of Gauss and Stokes which is derived from (B) by Euler's rule. This shows that V2F transforms into J.DV, where J2 a h g = 1. ! & b f \ \ 9 f c This result was given by Jacobi* with the foregoing derivation. The particular case in which was worked out by Lam6. The result is that 2 dV\ 3 / A8 8F This result is of great importance and will be used in the succeeding chapters to find potential functions and wave-functions which are simple functions of polar co-ordinates, cylindrical co-ordinates and other co- ordinates which form an orthogonal system. In the special case when dx2 -f dy* + dz* - *2 (r/£2 H- d^ -f d£2), Laplace's equation becomes a / dv\ a / dv\ a / sv\ ^^\K '^7 }^ ^ \K -*- H~ ^ K 3| V 9^ ; Srj\ drjj S and implies that /c-F is a solution of if K^ is a solution of this equation. Inversion is one transformation which satisfies the requirements, for in this case * = £/**, y = The inference is that if F (x, y, z) is a solution of Laplace's equation, the functiont , 1F x y z r \r2' r2' r is also a solution. Another transformation which satisfies the requirements is , _ ax_ r2 - a2 , _ r2 -f a2 ^ ~ y~+Tz ' ^ ~ 2 (y"+ " w) ' Z ~ 2i (y + iz) ' * Journ. ftir Math. vol. xxxvi, p. 113 (1848). See also J. Larmor, Caw6. Phil. Trans, vol. xii, p. 455 (1884); vol. xiv, p. 128 (1886). H. Hilton, Proc. London Math. Soc. (2), vol. xix, Records of Proceedings, vii (1921). Some very general transformation formulae are given by V. Volterra, Rend. Lined, ser. 4, vol. v, pp. 599, 630 (1889). f This result was given by Lord Kelvin in 1845. Special Transformations 161 In this case dx'* + dy'* + dz'* = - ^ --2 [dx* + <fy2 ~ a2 1 w) J and we have the result that if F (x, y, z) is a solution of Laplace's equation, ax r2j--_a2 r2 + ,-+-£ > 2 (*T-Mz) ' 2i is also a solution. These two results may be extended to Laplace's equation in a space of n dimensions ^y yy ^y dX}2 dx22 '" dxn2 If F (#!, #2, ... xn) is a solution of this equation, and if is also a solution*, and (*+ix\~*F\ T*-a2 r2t«8 ^3_ ^ 1 1 x "^ 2; L2 (»i + ^2) ' 2t (ij + tij J *! Hh is, ' ' ' ' ^ + ix J is a second solution. We shall now use this to obtain Brill's theorem. Putting Xj 4- ix2 = t, x± — ix2 = s, the differential equation becomes and the result is that if H2 = a;32 4- ... xn2, and if jP (s,t,x3, ... arn) is a solution, then n ~ is also a solution. Now a particular solution is given by 8 v = e u (t) x$, XD ... xn)) where U (t, #3, #4, ... xn) is a solution of the equation of heat conduction dU [d2U d2U d2 which is suitable for a space of n — 2 dimensions. The inference is that if U is one solution of this equation, the function a* ax3 ax, fy T"'*"~7 * The first result is given by B6cher, Bull. Amer. Math. Soc. vol. ix, p. 459 (1903). B II 162 Applications of the Integral Theorems of Gauss and Stokes is a second solution. When U is a constant the theorem gives us the particular solution which may be regarded as fundamental. EXAMPLES 1. Prove that if o - z - ct, p**x + iyt a «~z + ct, b ** x - iy, a'-z' + ctf, p-x' + iy', a' = z'-ct', b' - x' - iy , the relations a' (la — up — p) = — na 4- wp 4- r -f j3' ( — ma 4- v j3 4- g), a' (wz -f ft - e) = - tm - rib -f 0 - 0' (va 4- mb - /), a' (~ ma + v)8 -f q) - foz -f j£ -I- A; - 6' (Za ~ w/3 - ^p), a' (va -f- mb - f) = ja - M -f « + 6' (wa + Z6 - e), in which J, m, n, u, vt w,f, g, h, p, q, r, s, e,j, k are arbitrary constants, lead to a relation of type dxf* + dy'* -h dz'* - czdt'z » A2 (dx* -f- dy* -f ^22 - 2. Prove that if z - c* = ^ + (x + iy) 0, (z -f d) 0 = * - (x - iy), the relations of Ex. 1 give z' — ct' — $ + 0' (#' -f- iy'), 0' (*' + <*') = !/>' -(*'-»/), where X^ ^ w®' ~~ v<t>' + w^' 4- J, §2*51. jPAe equations for the equilibrium of an isotropic elastic solid. Let u, v. w be the components of the small displacement of a particle, originally at x, y, z, when a solid body is sliglitly deformed, and let X , Y, Z be the components of the body force per unit mass. We consider the variation of the integral / = Ldxdydz, where L = S — W, with vY 25 - (A + 2/i) (ux + vv 4- wz)* -f ^ [K + 'A and p being positive constants. The quantity 8 may be regarded as the strain energy per unit volume, while W is the work done by the body forces per unit volume. The densitj7 /> is supposed to be constant. We now wish L to be a minimum subject to the condition that the Isotropic Elastic Solid 163 values of u, v and w are specified at the boundary of the solid. The Eulerian equations of the "Calculus of Variations" give dz where Xx = 2pux + X(ux + vy + wz), Yz = Zy = ^ (ivy 4- vg), Yy = 2fjLVv 4- A (ux 4- vv 4- w,)9 Zx = .Yz = ^ (uz + wx)> Zz = 2/xtik 4- A (ux 4- vv + ^2), Xv = Yx = /x (t^ + wy). The quantities JT,,., yy, Zz, ya, Z^, Xv, are called the six components of stress, and the quantities exx*=ux) evv=vv, ezz=wz, eyz = wy + vz, eza. = ft, 4- w;x, e^y == vx 4- uV9 are called the six components of strain. In terms of these quantities 28 may be expressed in the form 28 = Xxexx 4- Yyevv 4- Zzc« + r,ev, 4- Zxegx 4- ^e^, while the relations between the components of stress and strain are Xx = 2fiexx 4- AA, Yz = Zy = Meyz, 7V - 2Metfv 4- AA, ZX = XZ = fieZX9 Zz = 2/iC« 4- AA, JCV = 7X = nexy, A = ux-\- vy + wz = exx + eyy 4- ezz . The relations may also be written in the form Eexx = Xx- v(Yy+ Zz), Eeyy = Yv - a (Zz + JQ, 1?€M = Z,-^ cr (Xx + Yv). The coelSicient E is Young's modulus, the number a is Poisson's ratio, and /A is the modulus of rigidity. The quantity A is the dilatation and — A the cubical compression. When xm = YV = zz = - ,, y2 = zx = xy = o, we have exx = eyy - ezz = - jp/(3A 4- 2/i), - A = !>/(A + hence the quantity k defined by the equation k = A + 164 Applications of the Integral Theorems of Gauss and Stokes is called the modulus of compression. The different elastic constants are connected by the equations F_M?\+2M) A *__J0_ A 4- /* ' 2~(A + /*)' 3 - 6(7* On account of the equations of equilibrium the expression for 87 may be written in the form (XxSu + YxSv + ZxSw) + | (Xy8u + Yybv + ZySw) r) 1 4- Sz (Xz8u 4- Yz8v + Z38w) dxdydz, and may be transformed into the surface integral [XvSu -f Yv where Xv = IX x -f- 4- ^2, . The quantities Xv, YV9 Zv are called the components of the surface traction across the tangent plane to the surface at a point under con- sideration. In many problems of the equilibrium of an elastic solid these quantities are specified and the expressions for the displacements are to be found. The equations of motion of an elastic solid may l3e obtained by re- d'^ii d^i) v^ijo garding - ^2- , — -^ , — ~ 2 as the components of an additional body force per unit mass. The equations are thus of type dXx dX 3XZ d2u § 2-52. The equations of motion of an inviscid fluid. Let us consider the variation of the integral / = I j \\Ldxdydzdt, where L = pl$ + a-+l (u* -f v» + w*)l +/(/>), <> < , and tt=^ + a3S v=? + a^, w=-J- + a-£ ....... (A) ox ox cy dy oz oz ^ ' Varying the quantities </», a, j3 and p in such a manner that the variations Vortex Motion 165 of </> and /? vanish on a boundary of the region of integration wherever particles of fluid cross this boundary, the Eulerian equations give - 0 ~"' , d d d ' 9 9 where - - == ^- + u 2- + v =-- + ti; ^ . a£ d< ox ofy dz If p = />/' (p) — / (p) it is readily seen that d^ __ I dp dv __ I dp dw __ 1 3p ~dt=z~~f>dx' dt = ~~'pdy' dt=~pdz' where f dp + || + « |£ + J (^ + & + l^2) = j^ ^ ....... (E) If £> is interpreted as the pressure, the last equation is the usual pressure equation of hydrodynamics for the case when there are no body forces acting. The quantities u, v, w are the component velocities and p is the density of the fluid at the point x, y, z. The equation (B) is the equation of continuity and the equations (D) the dynamical equations of motion. The relation p = />/' (p) — f (p) implies that the fluid is a so-called baro- tropic fluid in which the density is a function of the pressure. It should be noticed that with this expression for the pressure the formula for L becomes T ^ /A. L = F (t) - p when use is made of the relation (E). The foregoing analysis is an extension of that given by Clebsch*. The fact that L is closely related to the expression for the pressure recalls to memory some remarks made by R. Hargreaves| in his paper "A pressure integral as a kinetic potential." The equations of hydrodynamics may also be obtained by writing t+«d/t-l(u2+v2 + ">*)}+ f(p)> and varying <f>, a, j3, u, v, w and p independently. The equations (A) are then obtained by considering the variations of uy v and w. These equations give the following expressions for the com- ponents of vorticity : ~ ~ ~ , 0. . _ 9^ 9v __ 9 (a, j8) f~ 9i/~9z"9(*/,z)' _ du _ dw _ 9 (a, j8) ** ~ 97 " fa ~~ d(zyx) ' dv du 9 (a, j8) Crette'a Journ. vol. LVI (1859). t P^iL Mag. vol. xvr, p. 436 (1908). 166 Applications of the Integral Theorems of Gauss and Stokes These equations indicate that a = constant, j9 = constant are the equations of a vortex line. Now the equations (C) tell us that a and )3 remain constant during the motion of a particle of fluid, consequently a vortex line moves with the fluid and always contains the same particles. It should be noticed that in these variational problems no restrictions need be imposed on the small variations 80, 8/3 at a boundary which is not crossed by particles of the fluid because the integrated terms, derived by the integration by parts, vanish automatically at such a boundary of the region of integration on account of the equation which expresses that fluid particles once on the boundary remain on the boundary. § 2-53. The equations of vortex motion and Liouville's equation. Let us consider the variation of the integral (A) ' o x ' ' O " > " n / " \ » y fa d(x,y) the expressions for u and v being chosen so as to satisfy the equation of continuity, *« to = Q dx dy J for the two-dimensional motion of an incompressible fluid. Varying the integral by giving 0 and s arbitrary variations which vanish at the boundary of the region of integration, we obtain the two equations dxz dy2 dx \ dy) dy \ dx/ The first of these gives s = g (</r), where g (iff) is an arbitrary function which, when the region of integration extends to infinity, must be such that the integral / has a meaning. This requirement usually means that u, v and s must vanish at infinity. With the foregoing expression for s the second equation takes the form £t + !^+?W?'M==°> (B) which is no other than Lagrange's fundamental equation for two-dimen- sional steady vortex motion. In the special case when g ($) = Ae**, where A and h are constants, the equation becomes Liouville's Equation 167 This equation, which also occurs in Richardson's theory ot the space charge of electricity round a glowing wire*, has been solved by Liouvillef, the complete solution being given by where a and r are real functions of x and y defined by the equation a + ir = F (x + iy) and F (z) is an arbitrary function. Special forms of F which lead to useful results have been found by G. W. WalkerJ. In particular, if r* = x2 + y2, there is a solution of type -** — Mr I/TV f?Vl - - and when n = 1 the component velocities are given by the expressions 2i/ 2x ,„. , A = 2/ahy s = f-r - - - , 1 ' h(a* + r2) ' which are very like those for a line vortex but have the advantage that they do not become infinite at the origin. If we write ds ds ds 8 ~ -j* — u a~ + v 5~ > dt dx dy the quantity s may be defined by the equation s = — a tan"1 (y/x), and has a simple geometrical meaning. The quantity s may also be inter- preted as the velocity of an associated point on the circle r = a which is the locus of points at which the velocity is a maximum. It should be noticed that if we use the variational principle 3 f \(u2 + v2 - ^2) dxdy = 0, ...... (D) the corresponding equation is . ...... <E> and the solution corresponding to (C) is of type ~ __ 2y _ „__??__ U "" "" ^(02^72) ' V " A^a^-T2) ' This gives an infinite velocity on the circle r = a. * 0. W. Richardson, The Emission of Electricity from hot bodies, Longmans (1921), p. 50. The differential equation was formulated explicitly by ]Vf. v. La lie, Jahrbuch d. Radioaktivitat u. Elektronik, vol. xv, pp. 205, 257 (1918). f LiouviUe's Journal, vol. xvm, p. 71 (1853). { G. W. Walker, Proc. Roy. Soc. London, vol. xci, p. 410 (1915); BoUzmann Festschrift, p. 242 (1904). 168 Applications of the Integral Theorews of Gauss and Stokes Other solutions of (B) which give infinite velocities have been discussed by Brodetsky *. It seems that the variational principle (A) may have the advantage over (D) in giving solutions of greater physical interest. It should be noticed that if a boundary of the region of integration is a stream- line */r = constant, it is not necessary for 8s to be zero on this boundary. When the motion is in three dimensions an appropriate variation principle is 87 = 0, where / = £ n\(u*+v* + w*±$2) dxdydz, and the upper or lower sign is chosen according as the vortex motion is of the first or second type. To satisfy the equation of continuity when the fluid is incompressible and the density uniform, we may put 11 = a (<J' r) » = 9 ((7' r) m = d (°> r) 4 = 3 (*' a> r) 8(»,2)' d(z,x)' W d(x,y)' d(x,y,zy A set of equations of motion is now obtained by varying cr, r and s in such a way that their variations vanish on the boundary of the region of integration. These equations are 3_(5,CT, T) = () d (x, y, z) , Scr da da _ 9 («$, «s, cr) ' and are equivalent to the equations d d(s>^ which imply that These equations give du _ 1 3 jp dv _ 1 9j? rfit? _ 1 dp dt ~~ p dx ' dt~~ pdy* dt ~~ pdz* where the pressure y> is given by the equation ^ + \ (u2 -f v2 + w2 ± s2) = constant. The equation of continuity may also be derived from the variation problem by adopting Lagrange's method of the variable multiplier. In * S. Brodetsky, Proceedings of the International Congress for Applied Mechanics, p. 374 (Delft, 1924). Equilibrium of a Soap Film 169 ^\ ^\ ^\ this method / is modified by adding A( y +5-^4- ^ ) to the quantity within brackets in the integrand. The quantities A, u, v, w are then varied independently. It is better, however, to further modify / by an integration by parts of the added terms. The variation problem then reduces to the type already considered in § 2-52. § 2*54. The equilibrium of a soap film. The equilibrium of a soap film will be discussed here on the hypothesis that there is a certain type of surface energy of mechanical type associated with each element of the surface. This energy will be called the tension-energy and will be repre- sented by the integral !J TdS taken over the portion of surface under consideration, T being a constant, called the surface tension. This constant is not dependent in any way upon the shape and size of the film but it does depend upon the temperature. It should be emphasised that a soap film must be considered as having .two surfaces which are endowed with tension-energy. The tension-energy is not, moreover, the only type of surface energy; perhaps it would be better to say film energy ; for there is also a type of thermal energy associated with the film, and from the thermodynamical point of view it is generally necessary to consider the changes of both mechanical and thermal energy when the film is stretched. For mechanical purposes, however, useful results can be obtained by using the hypothesis that when a film stretched across a hole or attached to a wire is in equilibrium under the forces of tension alone, the total tension-energy is a minimum. Assuming, then, as our expression for the total tension -energy E E = 2T [ f (1 + zx2 4- z,2)* dxdy, the z-co-ordinate of a point on the surface or rim being regarded as a function of x and y, the Eulerian equation of the Calculus of Variations gives ~ ~ This is the differential equation of a minimal surface. When the film is subject to a difference of pressure on the two sides and the fluid on one side of the film is in a closed vessel whose pressure is pl while the pressure on the other side of the film is p2 , there is pressure- energy (pl ~ p2) V associated with the vessel closed by the film, where V is the volume of this vessel. Writing V in the form 170 Applications of the Integral Theorems of Gauss and Stokes where F0 is a constant and w is the perpendicular from the origin to the surface element dS, we consider the variation of the integral Now wH = z — xzx — yzy, and so the differential equation of the problem is 0 = ft - P* + 2 This differential equation may be interpreted by noting that the co- ordinates of a point on the normal at (x, y) are f = x-Rzx/H, 7) = y-Rzy/H, where R is the distance of the point from (x, y). If now two consecutive normals intersect at this point, we have 0 = d£ = dx - Ed (zx/H)y 0 - drj = dy - Rd (zv/H), for dR = 0. Expanding in the form 0 = dx [l - R A (z,///)] -dyB j- (zv/H), and eliminating dx, dy, we obtain as our equation for R n i P T^ (<> fff\^. ^ i 0 = l - R [dx (Z*IH) + dy ( If Rl and R2 are the roots, we have The quantities Rl and R2 are called the principal radii of curvature. A minimal surface is thus characterised by the equation -^ + p- = 0 and /h ^2 a surface of a soap film subject to a constant pressure-difference on its two sides is shaped in accordance with the equation —-_[---= constant. -«! -#2 When the film is subjected to only a smalLdifference of pressure and is stretched across a hole in a thin flat plate we can, to a sufficient approxi- mation, put H = 1 in this equation. The resulting equation is where K is a constant and the boundary condition is z — 0 on the rim. Torsion Problems and Soap Films 171 Now the same differential equation and boundary conditions occur in a number of physical problems and a soap-film method of solving such problems in engineering practice was suggested by Prandtl and has been much developed by A. A. Griffith and G. I. Taylor*. The most important problems of this type are : (1) The torsion of a prism ( Saint- Venant's theory). (2) The flow of a viscous liquid under pressure in a straight pipe. These problems will now be considered. EXAMPLES 1. The forces acting on the rim of a soap film of tension T are equivalent to a force F at the origin and a couple 0. Prove that 0= f%T[rx (nxds)}, where the vector ds denotes a directed element of the rim and the vector n is a unit vector along the normal to the surface of the film. Show by transforming these integrals into surface integrals that the force and couple are equivalent to a system of normal forces, the force normal to the element dS being of magnitude 2. The surface of a film closing up a vessel of volume V can be regarded as one of a family of surfaces for which C^ -f C2 is (7, a constant. If within a limited region of space there is just one surface of this family that can be associated with each point by some uniform rule and if S' is another surface through the rim of the hole, e the angle which this surface makes at a point (x, y, z) with the surface of the family through this point, the area of the outer surface of the film is I I cos c . dS'. Hence show that the area of the new surface is greater than that of the film if it encloses the same volume. 3. If w = zx, t> = zv, q***u* + ifi, show that the variation problem 8 ffo(q) dxdy = 0 leads to the partial differential equation where c2 [qQ" (q) - Of (q)] = q2 G' (q). Show also that the two-dimensional adiabatic irrotational flow of a compressible fluid leads to an equation of this type for the velocity potential z, the function G (q) being given by the equation O (q) - [2a2 + (y - 1) (U* - fW~\ where U, a and y are constants. * See ch. vn of the Mechanical Properties of Fluids (Blackie & Son, Ltd., 1923). 172 Applications of the Integral Theorems of Gauss and Stokes § 2-55. The torsion of a prism. Assuming that the material of the prism is isotropic, we take the axis of z in the direction of the generators of the surface and consider a distortion in which a point (#, y, z) is displaced to a new position (% + u, y + v, z + w), where u = — ryz, v = rzx, w = r</>, and </> is a function of x and y to be determined. The constant r is called the twist. This distortion is supposed to be produced by terminal couples applied in a suitable manner to the end faces. The portion of the surface generated by lines parallel to the axis of z (the mantle) is supposed to be free from stress. These are the simplifying assumptions of Saint-Venant. It is easily seen that du ___ dv __ dw _^dv du _ dx dy dz dx dy ~~ ' du dw /dd> p ____4_ __._,_ r ** dz^dx (dx dw dv p — _L — vz dy ^ dz ~ Hence, if Zx = fiezxy Zy = ^eys, Xx^ Yv = Zz = Xy == 0, the equations of equilibrium az? = 0 azy = 0 az az, = 0 dz ' 'dz ' dx dy show that Zx and Zy are independent of z and that 9 /dJ> \ d /d(j> \ >r U - 2/ + 3 - U + # ) = 0, dx \dx J ) dy \dy J 9¥ , 9V A d^+d^°- The boundary condition of no stress on the mantle gives IZX + mZv - 0, where (I, m, 0) are the direction cosines of the normal to the mantle at the point (x, y, z). Let us now introduce the function </f conjugate to </>, then 20 __ 3i/r d(f) __ 9i/f 3x dy ' 9?/ "~ 9x ' where r2 = a:2 + y2. The boundary condition may consequently be written in the form ~ ~ Torsion of a Prism 173 where % = iff — £r2 and ds is a linear element of the cross-section. This equation signifies that x is constant over the boundary and so the problem may be solved by determining a potential function i/j which is regular within the prism and which takes a value differing by a constant from \r- on the mantle of the prism. Without loss of generality this constant may be taken to be zero if there is only one mantle. It should be noticed that the function x satisfies the equation 32X , 92X__2 3x2 T dy2 and, with the above choice of the constant, is zero on the mantle when this is unique. It is often more convenient to work with the function x> especially as 9v - -" oy Since x vanishes on the mantle it is evident that \\Zxdxdy = 0, \\Zydxdy = 0. The tractions on a cross-section are thus statically equivalent to a couple about the axis of z of moment M = (.rZy - yZ,) dxdy = - pr * + y dxdy. Integrating by parts we find that M = The direction of the tangential traction (Zx, Zy) across the normal section of the prism by a plane z = constant is that of the tangent to the curve x — constant which passes through the point. The curves x = con- stant may thus be called "lines of shearing stress." The magnitude of the traction is LLT „-, where ~- is the derivative of v in a direction normal to r 3n on A the line of shearing stress. In the case of a circular prism x = i (a2 - r2), and* in the case of an elliptic prism where a and b are the semi-axes of the ellipse and 174 Applications of the Integral Theorems of Gauss and Stokes § 2-56. Flow of a viscous liquid along a straight tube. Consider the motion of the portion contained between the cross-sections z = zl and z == z1 -f h. If A is the area of the cross-section and /> the density of the fluid, the equation of motion is M* ^ = -4 (ft -A)- D, where pl and p2 are the pressures at the two sections and D is the total frictional drag at the curved surface of the tube. If u is the velocity of flow in the direction of the axis of z, u will be independent of z if the fluid is incompressible and so we may write u = u(x, y, t). We now introduce the hypothesis that there is a constant coefficient of viscosity p, such that where ~— denotes a differentiation in the direction of the normal to the on surface of the tube. Transforming the surface integral into a volume integral, we have the equation of motion A,du Al v , ffr /92^ 3^\ 7 7 PAh -ft = A (ft - ft) + J J V ^2 + gp) dxdy. Since h is arbitrary this may be written in the form Su dp - When the motion is steady this equation takes the form where — 2Jf£ = - >p and can be regarded as a constant, because u is in- dependent of z. This is the equation used by Stokes and Boussinesq. In the case of an elliptic tube where For an annular tube bounded by the cylinders r = a, r = b we may tt = JA: (a2 - r2) + JJP (62 - a2) log (6/r). The total flux $ is in this case n _ f6 , wJfa«f . (s2- I)2) , ,. . « - 2w Ja urdr - -4- r ~ log . r (5 = 6/a) Rectilinear Viscous Flow 175 and so the average velocity is 4 - If there is no pressure gradient the equation of variable flow is du where v = /x/p. This equation is the same as the equation of the conduction of heat in two dimensions. The fluid may be supposed in particular to lie above a plane z = 0 which has a prescribed motion, or to lie between two parallel planes with prescribed motions parallel to their surfaces. The simple type of steady motion of a viscous fluid which is given by the equation (I) does not always occur in practice. The experiments of Osborne Reynolds, Stanton and others have shown that when a viscous fluid flows through a straight pipe of circular section there is a certain critical velocity (which is not very definite) above which the flow becomes irregular or turbulent and is in no sense steady. From dimensional reason- ing it has been found advantageous to replace the idea of a critical velocity by that of a critical dimensionless quantity or Reynolds number formed from a velocity, a length and the kinematic viscosity v of the fluid. In the case of flow through a pipe the velocity V may be taken to be the mean velocity over the cross-section, the length, the diameter of the pipe (d). For steady "laminar flow" the ratio Vd/v must not exceed about 2300. In the case of the motion of air past a sphere a similar Reynolds number may be defined in which d is the diameter of the sphere. In order that the drag may be proportional to the velocity V the ratio Vd/v must be very small. EXAMPLES 1. In viscous flow between parallel planes x = ± a the velocity is given by an equation where c is the maximum velocity. Prove that the mean velocity is two-thirds of the maximum. 2. In a screw velocity pump the motion of the fluid is roughly comparable with that of a viscous liquid between two parallel planes one of which moves parallel to the other and drags the fluid along, although there is a pressure gradient resisting the flow. Calculate the efficiency of the pump and find when it .is greatest. Work out the distribution of velocity and the efficiency when the machine acts as a motor, that is, when the fluid is driven by the pressure and causes the motion of the upper plate. [Rowell and Finlayson, Engineering, vol. cxxvi. p. 249 (1928).] 3. The Eulerian equation associated with the variation problem 18 176 Applicatidns of the Integral Theorems of Gauss and Stokes L. Lichtenstein [Math. Ann. vol. LXIX, p. 514, 1910] has shown that when f(x, y) is merely continuous there may be a function u which makes 8/ = 0 and does not satisfy the Eulerian equation. § 2*57. The vibration of a membrane. Let T be the tension of the membrane in the state of equilibrium and w the small lateral displacement of a point of the membrane from the plane in which the membrane is situated when in a state of equilibrium, the vibrations which will be con- sidered are supposed to be so small that any change in area produced by the deflections w does not produce any appreciable percentage variation of T. The quantity T is thus treated as constant and the potential energy , (Sw\* , /3w\2l* , , + ( *-) + ( * dxdy \dxj \dyj J y is replaced by the approximate expression Let pdxdy be the mass of the element dxdy. The equation of motion of the membrane will be obtained by considering the variation of where „ Iff /dw\2 , j r7 IS = - \\ pi - I dxdy, V - The integral to be varied is thus = 2jJJ L^ \dt) where w -= 0 on the boundary curve for all values of t. The Eulerian equation of the Calculus of Variations gives where c2 = T/p. This is the equation of a vibrating membrane. The equation occurs also in electromagnetic theory and in the theory of sound. In the case when w is of the fotm . . ^ w == sin Ky . v (x, t), the function v satisfies the equation ^v . ri ~ K v 1x2 which is of the same form as the equation of telegraphy. It should be noticed that a corresponding variation principle m m , , , , o- -T\i~) -yhr) \dxdydzdt=*Q SxJ \dyj \3zjj y The Equation of Vibrations 111 gives rise to the familiar wave-equation d2w which governs the propagation of sound in a uniform medium and the propagation of electromagnetic waves. A function w which satisfies this equation is called a wave-function. Love has shown* that the equation (A) occurs in the theory of the propagation of a simple type of elastic wave. Taking the positive direction of the axis of z upwards and the axis of x in the direction of propagation, we assume that the transverse displace- ment v is given by the equation v = Y (z) cos (pt — fx). The components of stress across an area perpendicular to the axis of y are . V °v Y - n v — - x^^dx' -z*~u> z~^dz respectively and so the equation of motion *v dY, dY, , dY . (B) p'dt* " takes the form When p and JJL are constants this is the same as the equation of a vibrating membrane, but when p and //. are functions of z the equation is of a type which has been considered by Meissnerf. Transverse waves of this type have been called by JeffreysJ "Love waves," they are of some interest in connection with the interpretation of the surface waves which are observed after an earthquake. It may be mentioned that the general equation (B) may be obtained by considering the variation of the integral , Iff IT iSv J-2lJJKai and an extension can be made to the case in which p and JJL are functions of x, z and t. § 2-58. The electromagnetic equations. Consider the variation of the integral n\\ Ldxdydzdt, * Some Problems of Qeodynamics, p. 160 (Cambridge University Press, 1911). t Proceedings of the Second International Congress for Applied Mathematics {Zurich, 1926). { The Earth, p. 165 (Cambridge University Press, 1924). 178 Applications of the Integral Theorems of Gauss and Stokes where 2L = Hx* + Hv* + Ht* - E* - Ev* - E*, and H V -fc--fa, ^---g--^, (A) MV __ ?4* F = - ~A* " dx dy 9 * dt If the variations of Ax, AV) Az and 0 vanish at the boundary of the region of integration, the Eulerian equations give ~dy"~"dz ^"dT9 ~dz~~ 8^r=="aT5 ~dx ~ ~dy^~df9 dx dy dz In vector notation these equations may be written curlH=-~, divE=0, (B) and equations (A) take the form H=curlA, E«-~-V*. (C) ot These equations imply that !=-??, divH=0. (D) The two sets of equations (B) and (D) are the well-known equations of Maxwell for the propagation of electromagnetic waves in the ether; the unit of time has, however, been chosen so that the velocity of light is represented by unity. The foregoing analysis is due essentially to Larmor. Writing Q = H + iE, the two sets of equations may be combined into the single set of equations divQ=0. ...... (E) =-, ot By analogy with (C) we may seek a solution for which Q=-icurlL=-~?~VA. ...... (F) The relations between L and A may be satisfied by writing , --, ...... (G) Electromagnetic Equations 179 where G is a complex vector of type T + tH, while T and II are real ' Hertzian vectors ' whose components all satisfy the wave-equation When we differentiate to find an expression for Q in terms of G and K the terms involving K cancel and we find the Righi-Whittaker formulae H = curl(^curir-f ^), \ (I) E = curl (curl II - g If L = B + iA, A = Y -f- iO, where A, B, O and XF are real, we have ...... (J) E=-curlB=-8£- V<D, VI curir, B-- O = - div n, T = - div T, J where A, B, O and Y are wave-functions which are connected by the identical relations ^ ^^ = 0. ...... (L) The corresponding formulae for the case in which the unit of time is not chosen so that the velocity of light is unity are obtained from the foregoing by writing ct in place of t wherever t occurs. If we write Q' = eieQ, where 8 is a constant, it is evident that the vector Q' satisfies the same differential equations as Q and can therefore be used to specify an electromagnetic field (Ex, Hx) associated with the original field (E, H). It will be noticed that the function L' for this associated field is not the same as L, for L9 = H'2 - E"* = (H2 - E2) cos 20-2(E.H) sin 20. Also (W . H') = (H2 - E*) sin 20 + 2 (E . H ) cos 26. There are, however, certain quantities which are the same for the two fields. These quantities may be defined as follows : o JP u i? 17 n tox = &yJlz — &ZHV = Ux, XX = EX* + HX*-W, \ (M) Y, = EVE, + HVHZ = ZV,\ 180 Applications of the Integral Theorems of Gauss and Stokes It is interesting to note that these quantities may be arranged so as to form an orthogonal matrix * X iSx iSv We have, in fact, the relations x, Y, Z2 iS. w where T/f/2 C* 2 Q 2 C 2 T rr — &x — &y — &z ~ •*• i Xx Yx + AV Y y -f* -Xz •* 3 — GxGy = 0, V O i V O i V O i /"Y TI/ A Aa-Oa- + AVOV + A20Z + C^a; W = U, I=l(H*- E*)*+ (E.Hy2- .(N) § 2-59. TAe conservation of energy and momentum in an electromagnetic field. It follows from the field equations (B) and (D) that the sixteen com- ponents of the orthogonal matrix satisfy the equations PY ajr 8X, 8ga _ a^ 'dt ~~ J dx^ dy dx ' "^ i 57_v ayz 3y 3z rjiy ^^ C£jy Ofj^ dy dz dy dt d_G_> "dt dw ~dz = 0 = 0. .(A) Regarding Sx, Sy, Sz for the moment as the components of a vector S and using the suffix n to denote the component along the outward-drawn normal to a surface element da of a surface or, we have div/S.^T dW (dr — dxdydz) dt dr = - £ ^rfr. In this equation the region of integration is supposed to be such that the derivatives of 8 and W in which we are interested are continuous functions of x, y, z and t. This will certainly be the case if the field vectors and their first derivatives are continuous functions of x, y, z and t. * H. Minkowski, Gott. Nachr. (1908). Conservation of Energy and Momentum 181 Let us now regard W as the density of electromagnetic energy and S as a vector specifying the flow of energy, then the foregoing equation can be interpreted to mean that the energy gained or lost by the region en- closed by a is entirely accounted for by the flow of energy across the boundary. This is simply a statement of the Principle of the Conservation of Energy for the electromagnetic energy in the ether. The equations involving Ox , Gv , Gz may be regarded as expressing the Principle of the Conservation of Momentum. We shall, in fact, regard Gx as the density of the ^-component of electromagnetic momentum and (— Xx, — XV) — Xz) as the components of a vector specifying the flow of the ^-component of electromagnetic momentum. The vector S is generally called Poynting's vector as it was used to describe the energy changes by J. H. Poynting in 1884. The vector G was introduced into electromagnetic theory by Abraham and Poincare. In the case of an electrostatic field if there are only volume charges and the first integral is taken over all space, for then the surface integral may be taken over a sphere .of infinite radius and may be supposed to vanish when the total amount of electricity is finite and there is no electricity at infinity. It should be noticed that in the present system of units Poisson's equation takes the form V2</> -f p = 0, where p is the density of electricity. When there are charged surfaces an integral of type must be added to the right-hand side for each charged surface. The new expression for the total energy may be written in the form U = This may be derived from first principles if it is assumed that <f)8e is the work done in bringing up a small charge 8e from an infinite distance without disturbing other charges. Now suppose that each charge in an electrostatic field is built up gradually in this way and that when an inventory is taken at any time each carrier of charge has a charge equal to A times the final amount and a potential equal to A times the final 182 Applications of the Integral Theorems of Gauss and Stokes potential. A being the same for all carriers. As A increases from A to A -f dX the work done on the system is dU = 2 (Ac/>) ed\. Integrating with respect to A between 0 and 1 we get U = iSe</>. The carriers mentioned in the proof may be conducting surfaces capable (within limits) of holding any amount of electricity. If the carriers are taken to be atoms or molecules there is the difficulty that, according to experimental evidence, the charge associated with a carrier can only change by integral multiples of a certain elementary charge e. For this reason it seems preferable to start with the assumption that W represents the density of electromagnetic energy. On account of the symmetrical relations 72 = Zv,etc., Sx= O,, etc., we can supplement the relations (A) by six additional equations of types . (yZx -zYx) + (yZv -zYv) + (yZz -zY,)- (yG, - zGv) = 0, a (x8x + tX.) + (xSv + tXv) + - (x8. + tX.) + (xW - tO.) = 0. The equations of the first type may be supposed to express the Principle of the Conservation of Angular Momentum. We shall, in fact, regard yGz — zOy as the density of the ^-component of angular momentum and (zYx — yZx, zYv — yZy, zYz — yZz) as the components of a vector which specifies the flow of the ^-component of angular momentum. The equations of the second type are not so easily interpreted. We shall, however, regard xW — tOx as the density of the moment of electromagnetic energy with respect to the plane x = 0. This quantity is, in fact, analogous to Smo;, a quantity which occurs in the definition of the centre of mass of a system of particles. Here and in the relation S — G we have an indication of Einstein's relation (Energy) = (Mass) (square of the velocity of light) which is of such importance in the theory of relativity. The quantities (xSx -f tXX) xSv -f tXv, xSz -f tX9) will be regarded as the components of a vector which specifies the flow of the moment of electromagnetic energy with respect to the plane x = 0. The equation may, then, be interpreted to mean that there is conservation of the moment with respect to the plane x = 0. There is, in fact, a striking analogy with the well-known principle that the centre of mass of an isolated mechanical system remains fixed or moves uniformly along a straight line*. * A. Einstein, Ann. d. Physik (4), Bd. xx, S. 627 (1906); G. Herglotz, ibid. Bd. xxxvi, S. 493 (1911); E. Bessel Hagen, Math. Ann. Bd. ijcxxiv, S. 268 (1921). Conservation of Angular Momentum 183 EXAMPLES 1. Prove that when there are no external forces the equations of motion of an incom- pressible inviscid fluid of uniform density give the following equations which express the principles of the conservation of momentum and angular momentum, the motion being two- dimensional : - uy) - yp] + g- [pv(vx - uy) + xp] - 0. Hence show that the following integrals vanish when the contour of integration does not contain any singularities of the flow or any body which limits the flow, the motion being steady: I p (v -f iu) (id + vm) ds + I p (m + il) ds, I p (xv — yu) (ul + vm) ds -f I p (xm — yl) ds. When the contour does contain a body limiting the flow the integrals round the contour are equal to corresponding integrals round the contour of the body. 2. Let u (#! , x2 , ... xn) be a function which is to be determined by a variational principle 8/ =r 0, where r r r * I = II ... M(xl9x29 ...xnfu9ultu29 ...un)dxlidx^...dxn and ur = - - . Suppose further that 7 is unaltered in value by the continuous group of transformations whose infinitesimal transformation is (r = 1, 2, ... _ n and let Bu = Aw — 2 wrAxr, r-l — n dR then 08u- 2 ^T, r-i a*r where 5r - - f&xr - ^ 8u (r « 1, 2, ... w). When the function w satisfies the Eulerian equation 0 = 0 the foregoing result gives a set of equations of conservation. [E. Noether, Gott. Nachr. p. 238 (1918).] 3. If n = 2, / = (u^ — <w22 -h )3w2) ey*2 where o# )5 and y are arbitrary constants, we may write Aa^ = ea, A#2 = <2> ^w — — iv€2tt w^re ex and ea are two independent small quantities whose squares and products may be neglected. Hence show that the differential equation uu = aw22 + yaw2 4- /to leads to two equations of conservation A {2^1^ + V^} - (A + y) K2 + aW22 + ^w2 -f J5j {V + aw22 - /3w2} = (D2 -f where Z^EE /-, D2= =— . [E. T. Copson, Proc. ^rfm. Jfefa^. Soc. vol. XLII, p. 61 (1924).] dx^ 0X2 184 Applications of the Integral Theorems of Gauss and Stokes § 2-61. Kirchhoff' s formula. This theorem relates to the equation D2" + a (x, y, z, t) = 0, ...... (A) and to integrals of type u =-- I r~lf (t - r/c) F (XQ, y0, z0) dr0. Let us suppose that throughout a specified region of space and a specified interval of time, u and its differential coefficients of the first order are continuous functions of x, y, z and t ; let us suppose also that the differential d2u d2u coefficients of the second order such as ~ „ ; , ~ 2 and the quantity a are finite and integrable. Let Q be any point (x0, ?/0, z0) which need not be in the specified region of space and consider the function v derived from u by substituting t — r/c in place of t, r denoting the distance from Q of any point (x, y, z) in the specified region. It is easy to verify that v satisfies the partial differential equation __0 2r f 9 (x dv\ d /y Sv\ 3 /z where [a] denotes the function derived from a by substituting t — r/c in place of t. We now multiply the above equation by and integrate it throughout a volume lying entirely within the specified region of space. The volume integral can then be split into two parts, one of which can be transformed immediately into an integral taken over the boundary of this region. Let the point Q be outside the region of integration, then we have 3 /1\ I3v 2dr dv] ,0 f [a] , V ^ }- - * --- cf ^ \ dS + — dr' [_ dn\rj rdn crdndt] } r When Q lies within the region of integration the volume may be sup- posed to be bounded externally by a closed siirface S1 and internally by a small closed surface S2 surrounding the point Q. Passing to the limit by contracting 82 indefinitely the value of the integral taken over S2 is eventually ff f - \\\ JJ [_ wjjiere VQ denotes the value of v at Q and this is the same as the value of u at Q. Hence in this case fW^ flT a fl\ ldv 2dvdr']jv ±TtUQ == rfr - Vx - ( - ) -- a - - -^5~\dS. Q I r JJ l dn\rj rdn crdtdn] dv [du~] dr [dul Now Kirchhoff's Formula 185 hence finally we have Kirchhoff's formula* f M 7 ff (r n 8 /1\ 1 [dul 1 dr rSi^l) 7r> 4rrwQ = LJdr - N[w] =- (- - U- -- U, dS, Q 1 r jj \L *dn\r) r [dn] crdn[dt]\ where a square bracket [/] indicates that the quantity /is to be calculated at time t — r/c. When the point Q lies outside the region of integration the value of the integral is zero instead of UQ . When u and a are independent of t the formula becomes 47TUQ = l-dr — \\ \U~- (-} =- ,» , J r ]J(dn \rj rdn) and the equation for u is V*u + a (x, y, z) = 0. If we make the surface Sl recede to infinity on all sides the surface integrals can in many cases be made to vanish. We may suppose, for instance, that in distant regions of space the function u has been zero until some definite instant tQ. The time t — r/c then always falls below tQ when r is sufficiently large and so all the quantities in square brackets vanish. rs The surface integral also vanishes when u and ~ become zero at infinity and tend to zero as r -> oo in such a way that u is of order r~l and ~- , ~- of order r~2. In such cases we have the formula 477^— dr, (B) where the integral is extended over all the regions in which the integrand is different from zero. If [a] exists only within a number of finite regions which do not extend to infinity the function UQ defined by this integral possesses the property that UQ -> 0 like r0~l as r0 -> oo, r0 being the distance from the origin of co-ordinates, but it is not always true that -~ is of order r0~2. To satisfy this condition we may, however, suppose that ^- is zero for values of t *"*> — i less than some value t0 . Then if r is sufficiently large ^ is zero because — r/c falls below t0 . Wave-potentials of type (B) are called retarded potentials; the analysis shows that they satisfy the equation (A) and that the surface integral ffkilfiU-1^!-1*:^!! JJ ( Jdn\rJ r [dn] crdn[dt]\ * G.Kirchhoff, Berlin. Sitzungsber. S. 641 (1882); Wied. Ann. Bd. xvni (1883); Qes. AM. Bd. u, S. 22. The proof given in the text is due substantially to Beltrami, Rend. Lined (5), t. iv (1895), and is given in a paper by A. E. H. Love, Proc. London Math. Soc. (2), vol. i, p. 37 (1^03). An extension of Kirchhoff's formula which is applicable to a moving surface has been given recently by W. R. Morgans, Phil. Mag. (7), vol. ix, p. 141 (1930). 186 Applications of the Integral Theorems of Gauss and Stokes represents a solution of the wave-equation except for points on the surface S, for this integral survives when we put a = 0. It should be noticed, however, that when we put a = 0 the quantities [u] , «- , ^- become those relating to a wave-function u which is supposed in our analysis to exist and to satisfy the postulated conditions. When the quantities [u]> 21 > 12" are c"losen arbitrarily but in such a way that the surface integral exists it is not clear from the foregoing analysis that the surface integral represents a solution of the wave-equation. If, however, the quantities [u] , Ur- L ^r possess continuous second derivatives with respect to the time the integrand is a solution of the wave-equation for each point on the surface. It can, in fact, be written in the form where [u] -/*-, = ? ' ~ ' Now stands f or 7a 3 3 ZQ-+ W =- + 7&=- , d# <7J/ C7Z where I, m and ?i are constants as far as x, y and z are concerned and each term such as ^- -/(£ -- ) is a solution of the wave-equation; consequently OX \Jf \ C/ J the whole integrand is a solution of the wave-equation and it follows that the surface integral itself is a solution of the wave-equation. In the special case when a and u are independent of t we have the result that when or satisfies conditions sufficient to ensure the existence and finiteness of the second derivatives of V (see § 2' 32) the integral r is a solution of Poisson's equation V2F -f 47T(T (x, y, z) = 0, and the integral U = f [ \u J- (-} - - ^1 dS e JJ { dn\rj rdn) is a solution of Laplace's equation. § 2-62. Poisson's formula. When^he surface Sl is a sphere of radius ct with its centre at the point Q, [u] denotes the value of u at time t = 0 and Kirchhoff 's formula reduces to Poisson's formula * * The details of the transformation are given by A. E. H. Love, Proc. London Math. Soc. (2), vol. I, p. 37 (1903). Poisson's Formula 187 where /, g denote the mean values of /, g respectively over the surface of a sphere of radius ct having the point (xy y, z) as centre and u is a wave- function which satisfies the initial conditions u=f(x,y9z), ^ =0(z,2/,z), when t = 0. If we make use of the fact that each of the double integrals in Poisson's formula is an even function of t we may obtain the relation* ^ u(x,y,z,t)dt=f. This relation may be written in the more general form ^J u(x,y,z,8)ds 1 ffff2ir == j- u (x + CT sin 6 cos <f>, y -f CT sin 0 sin <f>, z -f CT cos 0, £) sin 9ddd<f>. 47T J 0 J 0 When w is independent of the time this equation reduces to Gauss's well- known theorem relating to the mean value of a potential function over a spherical surface. If u (x, y, z} s) is a periodic function of s of period 2r, where T is in- dependent of x, y and z, the function on the left-hand side is a solution of Laplace's equation, for if rt + r V = c2 u(x, y, z, s) ds, Jt-T we have V2F = c2 Vhids = ^ 9 d«9 = 0. It then follows that the double integral on the right-hand side is also a solution of Laplace's equation. If in Poisson's formula the functions / and g are independent of z the formula reduces to Parseval's formula for a cylindrical wave-function. Since we may write c2*2 sin Oddd<f> = da . sec 6 = ct (cH2 - />2)~* da, where da is an element of area in the #y-plane and p the distance of the centre of this element from the projection of the centre of the sphere, we find that 277 . u (x, y, *) = I JJ da . (cH* - p2)^/ (x 4- f , y 4- ij) da . (cW - p2)'* g (x + £, y + r?), where da = d^d-q and the integration extends over the interior of the circle 2 2 Cf. Rayleigh's Sound, Appendix. 188 Applications of the Integral Theorems of Gauss and Stokes This formula indicates that the propagation of cylindrical waves as specified by the equation G2w = 0 is essentially different in character from. that of the corresponding spherical waves. In the three-dimensional case the value of a wave-function u (x, y, z, t) at a point (#, y, z) at time t is fjfj completely determined by the values of u and ^- over a concentric sphere ot of radius cr at time t — T. If a disturbance is initially localised within a sphere of radius a then at time t the only points at which there is any disturbance are those situated between two concentric spheres of radii ct -f- a and ct — a respectively, for it is only in the case of such points that the sphere of radius ct with the point as centre will have a portion of its surface within the sphere of radius a. This means that the disturbance spreads out as if it were propagated by means of spherical waves travelling with velocity c and leaving no residual disturbance as they travel along. In the two-dimensional case, on the other hand, the value of u (x, y, t) at a point (x, y) at time t is not determined by the values of u and -^- over a concentric circle of radius CT at time t — T. To find u (x, y, t) we must f)tj know the values of u and -^- over a series of such circles in Which r varies ot from zero to some other value rl . If the initial disturbance at time t = 0 is located within a circle of radius a, all that we can say is that the disturbance at time t is located within a circle of radius ct + a and not simply within the region between two concentric circles of radii ct -j- a, ct — a respectively. Hence as waves travel from the initial region of disturbance with velocity c they leave a residual disturbance behind. The essential difference between the two cases may be attributed to the fact that in the three-dimensional case the wave-function for a source is of type r~lf (t — r/c), while in the two-dimensional case it is of type* CJ Jo L c This statement may be given a physical meaning by regarding the wave- function as the velocity potential for sound waves in a homogeneous atmosphere, a source being a small spherical surface which is pulsating uniformly in a radial direction. If /(0 = 0, (t<T0) = 1, (Tl>t>T0) = 0, (t>TJ, we have / \t — -cosh a \ da = 0, (ct < cTQ + />) Jo L c J - cosh-1 [c (t - T0)/p] , (cT0 + p < ct < cTl + p) - cosh-* [c (t - T0)//>] - cosh-i [c (t - Tj/p], (cTQ -h p < ct, cTi + p < ct). * Cf. H. Lamb, Hydrodynamics, 2nd ed. p. 474. Helmholtz's Formula 189 EXAMPLES 1. A wave-function u is required to satisfy the following initial conditions for t =» 0 u=f(x, y), -^ = 0 when 2 = 0, u = o, ™ = 0 when z ^ 0. C/I Prove that u is zero when z2 > c2t2 and when z2 < czt2 u = / where /denotes the mean value of the function / round that circle in the plane 2 = 0 whose points are at a distance ct from the point (x, y, 2). 2. If in Ex. 1 the plane z = 0 is replaced by the sphere r = a, where r2 = x2 + y2 -f z2, the wave- function w is equal to - / when there is a circle (on the sphere) whose points are all at distance ct from (x, y, z) and is otherwise zero. § 2-63. Helmholtz's formula. When a wave-function is a periodic function of t, Kirehhoff's formula may be replaced by the simpler formula of Helmholtz. Putting u = U (x, y, z) elkct the wave -equation gives V2C7 + k*U = 0. Applying Green's theorem to the space bounded by a surface S and a small sphere surrounding the point (xly yly zj we obtain formula (A) 477 U (x^y^zj =-*/ (*, y, *) (R~le~lkR) dS + fl-ic-** dS, where R* = (x - xtf + (y - yj* + (z - ^)2, and the normal is supposed to be drawn out of the space under considera- tion. This space can extend to infinity and the theorem still holds provided U -^ 0 like Ar~le~lkr as r-> oo, r being the distance of the point (x, y, z) from the origin. It is permissible, of course, for U to become zero more rapidly than this. A solution of the more general equation V2U + k*U + a> (x, y, z) - 0 is obtained by adding the term f 1 1 R-le~lkR a> (x, y, z) dxdydz to the right-hand side of (A) and it is chiefly in this case that we want to integrate over all space and obtain a formula in which U (xl9 yi, zj is represented by this last integral. In the two-dimensional case when u is independent of z, the function to be used in place of R-le~lkR is derived from the function *u= fit — -cosh a) da, Jo \ c / 190 Applications of the Integral Theorems of Gauss and Stokes already mentioned. Writing u = Ueikct as before, the elementary potential function satisfying V2U + k2U = 0 is K0 (ikp), where K0 (ikp) is defined by the equation K0(ikp) = e-ikKx»**da. Jo This is a function associated with the Bessel functions. For large values of R we have . . , while for small values of R KQ (iR) + log (B/2) is finite. The two-dimensional form of Green's theorem gives ...... (B) where p2 = (x - a^)2 + (y - 2/i)2, rf<§ is an element of the boundary curve and n denotes a normal drawn into the region in which the point (xl9 y^ is situated. A solution of the more general equation V2U + k*U + w (x, y) = 0 is likewise obtained by adding the term — J J K0 (ikp) w(x, y) dxdy to the right-hand side of (B). When k = 0 the corresponding theorem is that a solution of the equation V2t7 + eu (x, y) = 0 is given by 2-rrU = - I logpoj (x, y) dxdy. § 2-64. Volterra's method*. Let us consider the two-dimensional wave- equation r 3«« 9«» 3»« „ LWs-fc-te*-W~f(Xl*'Z) in which 2 = ct. If the problem is to determine the value of u at an arbitrary point (£, 77, £) from a knowledge of the values of -u and ite derivatives at points of a surface /S, we write Z=*s-£ r = 2/-7?, Z = z-f, and construct the characteristic cone X2 + Y2 = Z2 with its vertex at the point (£, 77, ^). We shall denote this point by P and the cone by the symbol F. * Ada Math. t. xvm, p. 161 (1894); Proc. London Math. Soc. (2), vol. u, p. 327 (1904); Lectures at Clark University, p. 38 (1912). Volterra's Method 191 Volterra's method is based on the fact that there is a solution of the wave-equation which depends only on the quantity ZjRy where R2 = X2 + 72. This solution, v, may, moreover, be chosen so that it is zero on the charac- teristic cone F. The solution may be found by integrating the fundamental solution (Z2 — X2 — Y2)~* with respect to Z and is cosh-1 w where w = Z/R. Since w = 1 on F it is easily seen that v = 0 on F. For this wave-equation the directions of the normal n and the co- normal v are connected by the equations cos (vx) — cos (nx), cos (vy) = cos (ny), »• cos (vz) — — cos (nz). At points of F the conormal is tangential to the surface and since v is ^77 zero on F, ~- is also zero. The function v is infinite, however, when R = 0 GV and a portion of this line lies in the region bounded by the cone F and the surface S. We shall exclude this line from our region of integration by means of a cylinder (7, of radius c, whose axis is the line R = 0. We now apply the appropriate form of Green's theorem, which is to the region outside C and within the realm bounded l->y S and F. On account of the equations satisfied by u and v the forgoing equation reduces simply to JU- *-*)"-- JJJ*fc- On C we have dS = and since lim (s log e) = 0 e ->0 we have Hm j j^ (u | - v^) - - 2» |* « tf , ,, «) &, where (^, r\> z) is on A$f and is in the part of 8 excluded by C. We thus obtain the formula and the value of u at P may be derived from this formula by differentiating with respect to £. The result is 192 Applications of the Integral Theorems of Gauss and Stokes EXAMPLE Prove that a solution of the equation dx* + By* = c2 W is given by the following generalisation of Kirchhoff's formula, 2nu (x, y,t) = / [c2 (t - tj* - r2]"* {cos nt - cos nr . c (t - tj/r} u fa, ft , fj dSl J <r f f in which r2 = (x — a^)2 + (y — ft)2 and the integration extends over the area a cut out on a surface 8 in the (x19yl9t1) space by the characteristic cone fa - *)2 + (ft - y)2 = c2 & - t)\ the time t being chosen so as to satisfy the inequality t < t± . [V. Voltorra.] § 2-71. Integral equations of electromagnetism. Let us consider a region of space in which for some range of values of t the components of the field-vectors E and H and their first derivatives are continuous functions of x, y, z and t. Take a closed surface S in this region and assign a time t to each point of S and the enclosed space in accordance with some arbitrary law t = f(z, y, z), where / is a function with continuous first derivatives. We shall suppose that this function gives for the chosen region a value of t lying within the assigned range and shall use the symbol T to denote the 'vector with 4. 3/ 3/ 9/ components^, 0^. Writing Q = H + iE as before we consider the integral / = Jj [Q - i (Q x T)]n dS taken over the closed surface S. The suffix n indicates the component along an outward -drawn normal of the vector P which is represented by the expression within square brackets. Transforming the surface integral into a volume integral we use the symbol Div P to denote the complete diver- gence when the fact is taken into consideration that P depends upon a time t which is itself a function of x, y and z. The symbol div Q, on the other hand, is used to denote the partial divergence when the fact that Q depends upon x, y and z through its dependence on t is ignored. We then have the equation / = 1 1 j div P . dr, where div P - div Q + T . R, 3Q and R - -57 — i curl Q. ot Integral Equations of Electromagnetism 193 Now div Q and R vanish on account of the electromagnetic equations and so these equations are expressed by the single equation / = 0. When / is constant T = 0 and the equation 7=0 gives which correspond to Gauss's theorem in magneto- and electrostatics. It may be recalled that Gauss's theorem is a direct consequence of the inverse square law for the radial electric or magnetic field strength due to an isolated pole. The contribution of a pole of strength e to an element EdS of the second integral is, in fact, eda>/4t7T, where cfa> is the elementary solid angle subtended by the surface element dS at this pole. On integrating over the surface it is seen that the contribution of the pole to the whole integral is e, \e or zero according as the pole lies within the surface, on the surface or outside the surface. This result is usually extended to the case of a volume distribution of electricity by a method of summation and in this case we have the equation where p denotes the volume density of electricity. Transforming the surface integral into a volume integral we have the equation J j J (div E - p) dr = 0, which gives div E = p. Since E = — V</> the last equation is equivalent to Poisson's equation V V + p = 0, in which the factor 4?r is absent because the electromagnetic equations have been written in terms of rational units. Our aim is now to find a suitable generalisation of this equation. In order to generalise Gauss's theorem the natural method would be to start from the field of a moving electric pole and to look for some generalisation of the idea of solid angle. This method, however, is not easy, so instead we shall allow ourselves to be guided by the principle of the conservation of electricity. The integral which must be chosen to replace P^T should be of such a nature that its different elements are associated with different electric charges when each element is different from zero. When the elements are associated with a series of different positions of the same group of charges which at one instant lie on a surface it may be called degenerate. In this case we B 13 194 Applications of the Integral Theorems of Gauss and Stokes can regard these charges as having a zero sum since a surface is of no thickness. Now it should be noticed that if we write 0 = *-/(*,*/, z), the quantity M SO , W ( 80 , W _ , ^ :/; = 27 + vx %- + vv 5- + v« o- = 1 — (0 • T) dt dt ox vcy z dz ' vanishes when the particles of electricity move so as to keep 6 = 0, that is so as to maintain the relation t = / (xy y, z), and in this case the integral dd is degenerate. We shall try then the following generalisation of Gauss's theorem and examine its consequences : I=i\\\P[l-(v.T)]dT. Transforming the surface integral into a volume integral we have 0 = [div Q - ip + T . (R 4- ipv)] dry and since the function / is arbitrary this equation gives div Q = ip, R = - ipv. Separating the real and imaginary parts we obtain the equations 3E curl H = -fa + />v, div E = p, curlE= - I?, divH=0, ct which are the fundamental equations of the theory of electrons. The first two equations give ~ which is analogous to the equation of continuity in hydrodynamics. Our hypothesis is compatible, then, with the principle of the conservation of electricity. The integral equation [Q- »(Q x T)]ndS= »}J|p[l - (v.T)]dr (A) will be regarded as more fundamental than the differential equations of the theory of electrons if the volume integral is interpreted as the total charge associated with the volume and is replaced by a summation when .the charges are discrete. This fundamental equation may be used to obtain the boundary conditions to be satisfied at a moving surface of discontinuity which does not carry electric charges. Let t = f (x, y, z) be the equation of the moving surface and let the Boundary Conditions 195 surface 8 be a thin biscuit-shaped surface surrounding a superficial cap S0 at points of which t is assigned according to the law t =/(#, y, z). At points of the surface S we shall suppose t to be assigned by a slightly different law t = /x (#, t/, z) which is chosen in such a way that the points of S on one face have just not been reached by the moving surface t = f (x, y, z), while the points on the other face have just been passed over by this surface. Taking the areas of these faces to be small and the thick- ness of the biscuit quite negligible the equation (A) gives [Q' - • (Q' x T)]» = [Q" - i (Q" x T)]n, where Q', Q" are the values of Q on the two sides of the surface of dis- continuity and the difference between /x and / has been ignored. Writing q = Q' — Q" we have the equation [q - * (q x T)]n = 0. Now the direction of the cap $0 is arbitrary and so q must satisfy the relation Q-.'(qxT). This gives q2 = 0. Hence if q = h + ie we have the relations h* - & = 0, (h.e) = 0. The equation also gives (q . T) = 0, and (q x T) = i (q x T) x T - i [T (q . T) - qT2] = - iT2q or T2 = 1 if q ^ 0. Hence the moving surface travels with the velocity of light. A similar method may be used to find the boundary conditions at the surface of separation between two different media. We shall suppose that the media are dielectrics whose physical properties are in each case specified by a dielectric constant K and a magnetic permeability /*. For such a medium Maxwell's equations are , divD-0, ^-— where D = #E, B Instead of these equations we may adopt the more fundamental integral equations J| [B + (E : which give the generalisations of Gauss's theorem. The boundary con- ditions derived from these equations by the foregoing method are d - (h x T) = 0, b + (e x T) = 0, 13-2 196 Applications of the Integral Theorems of Gauss and Stokes where e, h, d, b are the differences between the two values of the vectors E, H, D, B respectively on the two sides of the moving surface. These equations give (A.T) = 0, (b.T)^0, (d x, T) + T*h = (h . T)T; (b x T) - T2e - - (e . T) T. If the vector v represents the velocity along the normal of the moving surface we have v^ ^ y VT = \ hence the equations may be written in the form dv = 0, bv = 0, hr = (v y d)r, eT = — (v x b)T, where dv , bv denote components of d and 6 normal to the moving surface and the suffix r is used to denote a component in any direction tangential to the moving surface. When this surface is stationary the conditions take the simple form dv = 0, b¥ = 0, hr = 0, er = 0 used by Maxwell, Rayleigb and Lorentz. When a surface of discontinuity moves in a medium with the physical constants K and /z, we have Heaviside's equations (Electrical Papers, vol. ir, p. 405) X(e.T) = 0, /*<h.T) = 0, K (e x T) + T2h - 0, p. (h x T) - T2e = 0,. and so A> [(h x T) x T] = KT* (e x T) - - T4h, i.e. Kfji - T2 if h ^ 0. The surface thus moves with a velocity v given by the equation § 2-72. The retarded potentials of electromagnetic theory. The electron equations , ~™ i [ (j ty \ curl H = - { ~ + pv), div E = p, C \ ut I curl E = - -37, div // = 0 c 3£ may be satisfied by writing c o£ where the potentials A and O satisfy the relations Retarded Potentials 197 The last equations are of the type to which Kirchhoff 's formula is applicable and so we may write ...... (B) These are the retarded potentials of L. Lorenz. The corresponding potentials for a moving electric pole were obtained by Lienard and Wiechert. They are similar to the above potentials except that the quantity — c/M of § 1-93 takes the place of 1/r. Let £ (£), ^ (t), £ (t) be the co-ordinates of the electric pole at time t and let a time r be associated with the space-time point (x, y, z, t) by means of the relations [X - t (T)]« +[y-r, (T)]« + [z - £ (T)]« = c* (t - T)«, r < t, (C) then M=[x-t (r)} ? (r) + [y-r, (r)]r,' (r) + [z - I (r)] £' (r) - c' (« - r), and if e is the electric charge associated with the pole the expressions for the potentials are respectively A --*>(?\ A -_?9l<T) A -_<» *-_ e-C x~ '' ~ ' z~ ' These satisfy the relation (A) and give the formulae of Hargreaves ,-, e d (or, T) „ _ e 9 (a, r) x = 4^c 3 (x, 0 ' * = ITT 3ly, z) ' where o = [* - f (r)] f' (r) + [y - i, (T)] ," (T) + [z - £ (T)] C" (T) + C* - [£' (T)]« - [,' (T)]» - [£' (T)]«. It should be remarked that the retarded potentials (B) can be derived from the Lienard potentials by a process of integration analogous to that by which the potential function == ttl is derived from the potential of an electric pole. Instead of considering each electric pole within a small element of volume at its own retarded time r we wish to consider all these electric poles at the same retarded time TO belonging, say, to some particular pole (&> >?o> £o, TO)- Writing f W = £> (a) + a (a), rj (a) = rjQ (a) + ft (a), £ (or) = £0 (a) + y (a), where a (a), ft (a), y (a) are small quantities, we find that if T is defined by (C), (r - T0) [(x - &) & +(y- %) V + (2 - Co) &,' - c« (« - T0)] + (*-&)«+(»- %) ^ + (z - £0) 7 = 0, where £0, ij0, £0, ^0'> ^o'> £o'> a> /S> y are all calculated in this equation at time TO. 198 Applications of the Integral Theorems of Gauss and Stokes On account of the motion the pole (£, 77, £) occupies at time r the position given by the co-ordinates £ = fo + « + (r - T0) &', *? = i?o + j8 + (T - T0) V. >a £ = £o + y + (r - T0) £0'. -1^'* If p is the density of electricity when each particle in an element of volume is considered at the associated time T and p0 is the density when each particle is considered at time TO, we have pd (f, T?, £) - Pod (a, j8, y). C7" Therefore o0 == p — -- , ru r cr — (r . v) and so f I f ^ dxdydz - f f f -^- , dxdydz. c JJJ ^ JJJ^-(^.V) Writing p dxdydz ^ de we obtain the Lienard potentials. Similar analysis may be used to find the field of a dipole which moves in an arbitrary manner with a velocity less than c and at the same time changes its moment both in magnitude and direction. Let us consider two electric poles which move along the two neigh- bouring curves * = £(*), y-^W, * = £(0> x = f(t)+€a (t), y = TJ (*) + cjB (t), z = J (0 + ey (0, e being a quantity whose square may be neglected. If TX is defined in terms of xy y, Zy t by the equation and T! = T + €0, we easily find that JM + a (T) [x-£ (T)] + ft (r) [y _ , (T)] + y (T) [z _ J (T)] = Q. If 3/x is the quantity corresponding to M , we have J^ = Jf + c [^^CT + (x - f)a' + (y - T?) $ -f (^ - t) / - «f - fa' - y£'] = Jlf -f €[&Jf<7 + p], say, where a has the same meaning as before and primes denote differentia- tions with respect to T. Now if _ eff'frJ + ^Kll ,.,,_ ec ' ~ -_ 477J/' Moving Electric and Magnetic Dipoles 199 we have ax = [A.' -Ax}=- , [Ma' - p? + MB?' - MOaf], But Ma! - p? + M6(" - M6a£' s (y - r,) n' - (z - Q m' - c2 (t - T) «' + c2« - nrf + m£' + o{a(t- T) — n(y — ij) + m(z — £)}» where I = ft' - m', m =* y? - oj', » = <nj' - #'. Hence we may write _ e riL^7^ _ i f!M 0.1 f-^i a* ~ ~ 47r [fy \M) dz \MJ + 9< \M)\ ' ecp /«\ a //?\ a/y\i ^ ~ 4^ [ai UJ + % \M) + dz \M)\ • These results may be obtained also with the aid of the general theorem which gives the effect of an operation -r analogous to differentiation, i r i dn = i n ^ i dn = i n ^DI , i n i f drj ao; [M d (e, r) J " dy lMd(€,r / being a function of r and 6. Writing / = f , ^- = a, =-' = /J, ^- = y the expression for a^ is at once obtained from that for ^4X. Writing/ = r we obtain the expression for (f>. The formulae show that the field of the moving dipole may be derived from Hertzian vectors II and F by means of the formulae where u and w are vector functions of r with components (a, j3, y), (Z, m, n) respectively. If v denotes the vector with components (£', T\ ', £x) we have the relations (v . w) = 0, (u . W) - 0, consequently Hertzian vectors of types (D) do not specify the field of a moving electric dipole unless these relations are satisfied. Since A = - 37 + curl T, B = - ^ - curl II, c ot c ot where B and O are the electromagnetic potentials of magnetic type, we may write down the potentials for a moving magnetic dipole by analogy. 200 Applications of the Integral Theorems of Gauss and Stokes We simply replace II by T and F by — II. Hence the potentials of a moving magnetic dipole are of type a - - -- x ~ 4n dy \M) dz \M) dt \M me f" 3 / / \ 9 fm\ d / n ~ Let us now calculate the rate of radiation from a stationary electric dipole whose moment varies periodically. Taking the origin at the centre of the dipole, we write HX = ^/(T), Hv=±g(T)9 ilz = -rh(r), r=«-r/c, where /, y, h are periodic functions with period T. The vector is zero since there is no velocity. In calculating the radiation we need only retain terms of order 1/r in the expressions for E and H . To this order of approximation we have =- (yE, - zEy), where ^ = ^ (/"* + g"* + h"*) - ~ (xf" + yg" + zh")*. The rate of radiation is obtained by integrating cE2 over a spherical surface r = a, where a is very large. With a suitable choice of the axis of z we may write and the value of the integral over the sphere is the mean value over a period T is A2 /27T\4 " " Electromagnetic Radiation 201 EXAMPLE and L (x, y, z, t, s) = [x - £ (s)] I (s) + [y — ^ (5)] m (*) -f [2 - £ («*)] n (s) - c(t ~ s), prove that the potentials 1 iT ft i \ l(s)ds v£'(T) * = 4w J - oo £ (#, y, 2, /,~«) " ' ( 4^ ' 4 _ 1 fr ///^ mW^ / ^' £(*,*,,*,*,*) 'V''4^' are wave-functions satisfying the condition 1V "*" c dt = * Show also that in the field derived from these potentials the charge associated with the moving point £(T), y (T), £(T) is/(r), the variation with the time being caused by the radiation of electric charges from the moving singularity in a varying direction specified by the direction cosines / (r), m (T), n (T). § 2-73. The reciprocal theorem of wireless telegraphy. If we multiply the electromagnetic equations JSj = — curl (cEi), C1 = curl (c^) for a field (Ely H^ by H2, — E2 respectively, where (E2, H2) are the field vectors of a second field in the same medium, and multiply the field equations A , , ^ x ~ i / r? x ^ B2 = — curl (cE2), C2 = curl (cH2) for this second field by — H19 El respectively and then add all our equations together, we obtain an equation which may be written in the form (H2 . B,) - (H, . B2) + (El . C2) - (E2 . C,) = cdiv (E2 x HJ - cdiv (EI x H2). (A) We now assume that both fields are periodic and have the same frequency co/277. Introducing the symbol T for the time factor e~ltat and assuming that it is understood that only the real part of any complex expression in an equation is retained, we may write TT WL u nnjt ~& ^n^t & nn^i /i-i==j[/ij,. juL2 — j. n2j /vj = JL 6j , jG/2 — ,JL e2) where the vectors ely hl9 cl9 etc. depend only on x, y and z. Now let K, fji and a be the specific inductive capacity, permeability and conductivity of the medium at the point (x, y, z) and let a denote the quantity (ioo -f icr)/c, then we have the equations B2 == — ia)jjih2T, Cl = — iceiTa, C2 = — ice2Ta, 202 Applications of the Integral Theorems of Gauss and Stokes which indicate that the left-hand side of equation (A) vanishes. The equation thus reduces to the simple form * div (e2 x hj) = div (e1 x h2). This equation may be supposed to hold for the whole of the medium surrounding two antennae f if these sources of radiation are excluded by small spheres Kl and K2 . An application of Green's theorem then gives [ (% x ^i)n dS -f [ (e2x hi)n dS - I (el x h2)n dS + [ (el x h2)n dS, J KI J Kt J KL J KZ an equation which may be written briefly in the form Jn -f- J21 — J12 4- </22- Let the first antenna be at the origin of co-ordinates and let us suppose for simplicity that it is an electrical antenna whose radiation may be represented approximately in the immediate neighbourhood of O by the field derived from a Hertzian vector TIT with a single component HZT, where = MelpR (c2p2 = kfjLa>2 -f and M is the moment of the dipole. In making this assumption we assume that the primary action of the source preponderates over the secondary actions arising from waves reflected or diffracted by the homogeneities of the surrounding medium. Using % to denote the value of a at 0 and writing FT for 3FI/9r, (£2 > y* > £2) for the components of e2 , we have n = - ia, f {(** J K, -f y2) £2 - a*& - yz^} H'R~*dS. For the integration over the surface Kl the quantities 7?2, Ilx and the vector e2 may be treated as constants, for K1 is very small and e2 varies continuously in the neighbourhood of 0. We also have [yzdS - \zxdS = \xydS = 0, (x*dS= ^dS= \z2dS = IxdS^O, lx*dS = 0, l Therefore Jn = - and, since lim .R->0 we have finally Ju = 2ia1M1£2/3. Now the rate at which the antenna at 0 radiates energy is 8 = ^W2127rF3, V = c (€x)'i. * H. A. Lorentz, Amsterdam. Akad. vol. iv, p. 176 (1895-6). f A. Sommerfeld, Jahrb. d. drahtl Telegraphie, Bd. xxvi, S. 93 (1925); W. Schottky, ibid. Bd. xxvn, S. 131 (1926). Reciprocal Relations 203 Assuming that 8 is the same for both antennae we obtain the useful expression The integral J21 is seen to be zero because it involves only terms which change sign when the signs of x, y and z are changed. Evaluating «/22 and Ju in a similar way we obtain the equation (^ + icrjaj) (VtfK^ £2 = (ic, + iaju) (F23//c2)i &, where f1? rjl9 £x are the components, at the second antenna 02, of the vector elf The amplitudes and phases of the field strength received at the two antennae are thus the same when both antennae are of the electric type and are situated at places where the medium has the same properties and emits energy at the same rate. When the two antennae are both of magnetic type the corresponding relation is (M2F23)*ri=(Mi^3)*y2, where a^ , & , y^ are the components of ht and a2 , /J2 , y2 are the components of A2. The antennae are again supposed to be directed along the axis of z but there is a more general theorem in which the two antennae have arbitrary directions. The relation (A) and the associated reciprocal relation remind one of the very general extension of Green's theorem which was given by Volterra* for the case of a set of partial differential equations associated with a variational principle. This extension of Green's theorem is closely connected with a property of self-adjointness which has been shown by Hirsch, Kiirsch&k, Davis and La Pazj to be characteristic of certain equa- tions associated with a variational principle. In the case of the Eulerian equation F = 0 associated with a variational principle 87 = 0, where l>i,32, ••• Xn\u\ ultu2j ... un]dxldx2 ...dxn, du d*u . 1 0 Us = 7^ > u" = ^T^r> (r> s = !> 2> ••• n)> x UJsg , CJCr VJUg the equation which is self -adjoint is the "equation of variation" for v dF 3F dF dF 3F 0 = v = -- h ^ -A -- h ... vn ~ -- h vn -- --- h v12 x- - + ... . du 1 dut dun dun li du12 * V. Volterra, Rend. Lincei (4), t. vi, p. 43 (1890). f A. Hirsch, Math. Ann. Bd. XLIX, S. 49 (1897); J. Kurschdk, ibid. Bd. LX, S. 157 (1905); D. R. Davis, Trans. Amer. Math. Soc. vol. xxx, p. 710 (1928); L. La Paz, ibid. vol. xxxir, p. 509 (1930). CHAPTER ^ TWO-DIMENSIONAL PROBLEMS § 3-11. Simple solutions and methods of generalisation of solutions. A simple solution of a linear partial differential equation of the homo- geneous type is one which can be expressed in the form of a product of a number of functions each of which has one of the independent variables as its argument. Thus Laplace's equation 927 == dx2 dy2 possesses a simple solution of type V = e~my cos m(x - x'), ...... (A) where m and x' are arbitrary constants ; the equation 9F_ d2V ~ft-K~dx* ppssesses the simple solution V = e~Km2t cos m (x - x'}, ...... (B) and the wave-equation ^v dx2 ^ c2 possesses the simple solution V = — sin met cos m (x — x'). ...... (C) *• itv The last one is of great historical interest because it was used by Brook Taylor in a discussion of the transverse vibrations of a fine string. It should be noticed that the end conditions F = 0 when x = ± a/2 are satisfied by a solution of this type only if ma = 2n -f 1, where n is an integer. There are thus periodic solutions of period T = 27r/wc - 2ira/(2n -f 1). If M (m., x') denotes one of these simple solutions a more general solution may be obtained by multiplying by an arbitrary function of m and x' and then summing or integrating with respect to the parameters m and x'. This method of superposition is legitimate because the partial differential equations are linear. When infinite series and infinite ranges of integration are used it is not quite evident that the resulting expression will be a solution of the appropriate partial differential equation and some Generalisation of Simple Solutions 205 process of verification is necessary. If, for instance, we take as our generalisa- tion the integral V = f°° M (m, x')f(m] dm (t > 0, y > 0), Jo and distinguish between solutions of the different equations by writing v for V when we are dealing with a solution of the second equation and y for V when we are dealing with a solution of the third equation, we easily find that when / (m) = 1 we have 2v = (77//c£)*exp [— (x — x')*l±Kt], y = - , - or 0 according as | x — x' \ = c£ (£ > 0). It is easily verified that these expressions are indeed solutions of their respective equations. These solutions are of fundamental importance because each one has a simple type of point of discontinuity. In the last case the points of discontinuity for y move with constant velocity c. We may generalise each of these particular solutions by writing V, v or y equal to M (m, x') F (xf) dx'dm, Jo J -oo where the integration with regard to x' precedes that with respect to m. When the order of integration can be changed without altering the value of the repeated integral the resulting expressions are respectively F=p° yF(x')dx' 2v = (77/Ac*)* I"" exp [- (x - x')*l±K(\ F (xf) dx', J -00 — fjr + ct y=l\ F(x')dx'. * Jx-ct The last expression evidently satisfies the differential equation when F (x) is a function with a continuous derivative; y represents, moreover, a solution which satisfies the conditions when t = 0. When the function F (x) is of a suitable type the functions V and v also satisfy simple boundary conditions. This may be seen by writing x' - x + y tan (6/2) in the first integral and x' =* x + 2u (irf)* 206 Two-dimensional Problems in the second. The resulting classical formulae Too v = TT* F[x+2u (AC J -00 u suggest that V = nF (x) when t/ = 0 and v = -nF (x) when 2=0. These results are certainly true when the function F (x) is continuous and integrable over the infinite range but require careful proof. The theorems suggest that in many cases * rrF (x) = [ °° dm f °° cos m(x- x') F (x') dx'. Jo J -oo This is a relation of very great importance which is known as Fourier's integral theorem. Much work has been done to determine the conditions under which the theorem is valid. A useful equivalent formula is (x) = I °° dm f °° etm(*-*'> F (xf) dx'. J —oo J —oo When F (x) is an even function of x Fourier's integral theorem may be replaced by the reciprocal formulae f 00 F (x) — cos mxG (m) dm, Jo 2 f °° 0 (m) =- co$mxF(x)dx, 7T Jo and when F (x) is an odd function of x the theorem may be replaced by the reciprocal formulae f °° F (x) = sin mxH (m) dm, Jo 2 f°° H (m) = - sin mxF (x) dx. TT JO The formulae require modification at a point x, where F (x) is discon- tinuous. It F (x) approaches different finite values from different sides of * The theorem is usually established for a continuous function which is of bounded variation fee A) and is such that / | F (x) \ dx and I | F (x) \ dx exist. F (x) may also have a finite number of / -00 J -JO points of discontinuity at which F (x + 0) and F (x - 0) exist but in this case the integral represents 'J[[F(x+ 0)+ F (x - 0)]. Proofs of the theorem are given in Carslaw's Fourier Series and 2 • Integrals; in Whittaker and Watson's Modern Analysis; and in Hobson's Functions of a Eeal Variable. Fourier' '$ Inversion Formulae 207 the point x the integral is found to be equal to the mean of these values instead of one of them. Thus in the last pair of formulae we can have F(x)=l x«f>, H(m)= 2-[l-cosw], 77 = 0 X> (f>, but the integral gives F (1) = \ . EXAMPLE If S (x, t) = (7r/c<)~i exp [- x*/4:Kt] and / (x) is continuous bit by bit a solution of %r- = K x-2, which satisfies the condition y = f (x) when < = 0 and — oo < jc < oo , is given by the formula v = 4 T flf (x - *0, 0/(s0) ^o + 4 3 [/ (*n) - f(xn)] J — oo n=l x /J [S (a? -a;n -£,«)- fl (* - s,, + £, *)] <fc where 2/ (#„) = / (zn + 0) + / (xn — 0) and the summation extends over all the points of discontinuity of/ (x). § 3-12. A study of Fourier's inversion formula. The first step is to establish the Biemann-Lebesgue lemmas*. Let g (x) be integrable in the Riemann sense in the interval a < x < b and when the integral is improper let | g (x) \ be integrable. We shall prove that in these circumstances f b lim sin (kx) . g (x) dx = 0. k ->w J a Let us first consider the case when g (x) is bounded in the range (a, 6) and G is the upper bound of \g (x)\. We divide the range (a, 6) into n parts by the points #15 x2, ... xn_! and form the sums Sn = U, (x, - a) + U2 (x2 - x,) + ... Un (b - xn^)9 sn — L± (Xi ~ a) 4- L2 (x2 — x^ -f ... Ln (b — xn_j), where C7r, Lr are the bounds of g (x) in the interval serHl < x < xr, so that in this interval Since & (x) is integrable we may choose n so large that Sn — sn < c, where c is any small positive quantity given in advance. Now g (x) sin Icxdx = S gr (xr^) sin kx . dx + 2 ojr (x) sin kx ' dx * The proof in the text is due to Prof. G. H. Hardy and is based upon that in Whittaker and Watson's Modern Analysis. 208 Two-dimensional Problems the summations on the right being from r = 1 to r = n and the integrations from #r_, to xr. With the same convention r b I (b g (x) sin kxdx I Ja sin kx . dx + < 2*167* -f Sn - sn < 2nG/k -f c. Keeping ?i fixed after e has been chosen and making k sufficiently large we can make the last expression less than 2e and so the theorem follows for the case in which g (x) is bounded in (a, b). When g (x) is unbounded and | g (x) \ Integra ble in (a, b) we may, by the definition of the improper integral, enclose the points at which g (x) is unbounded in a finite number of intervals il9 i2, ... ip such that P S I g (x) I dx < €. r~lJi, Now let G denote the upper bound of g (x) for values of x outside the intervals ir and let e1? <?2, ... ep+l denote the portions of the interval (a, b) which do not belong to ilt i2> ... iv, then we may prove as before that g (x) sin kx . dx g (x) sin kx . dx P r I -f- 2 \ g (x) sin kx dx < 2nG/k 4- 2e. Now the choice of e fixes n and 6r, consequently the last expression may be made less than 3e by taking a sufficiently large value of k. Hence the result follows also when g (x) is unbounded, but subject to the above restriction. Some restriction of this type is necessary because in the case* when g (x) is the unbounded function x~l (1 — x2)"* for which | g (x) \ is notintegrable in the range (— 1, 1) we have j-l fA: sin kx g (x) dx '= TT J0 (r) rfr, J - 1 Jo r A r co and as k -> oo J0 (r) dr = JQ (r) dr = 1. Jo Jo The next step is to show that if x is an internal point of the interval (—a, /?), where a and /J are positive, and if / (x) satisfies in (—a, j8) the following conditions : (1) f (x) is continuous except at a finite number of points of dis- continuity, and if / (x) has an improper integral | / (x) \ is integrable ; (2) / (x) is of bounded variation, then limf 1C -^oo J - a Let us write sin k (t — x) [x - 0)] = ^ - L +i: say. Fourier's Integrals 209 and transform the integrals by the substitutions t = x — u and t = x -f- u respectively, then r/3 r 55*( J — a & — fa Jo -f- f 0 — J? ai ^ Jo fa-t-jr _^)_/(z_0)]c^4-/(z- 0) sin ku. du/u Jo r&-x « + ^) -/(* + Q)]du+f(x+ 0) sinfai.dWtt Jo Now let c denote one of the two positive quantities a -f x, j3 — x, then fc r l:c n sin ku . du/u = sin v . dv/v -> ^ as fc -> oo. Jo Jo * Also, let F (u) denote one of the two functions / (x — u) - / (x — 0), / (x + w) — / (x -j- 0), then jP (0) = 0 and .F (u) is of bounded variation in the interval (0 < u < c). We may therefore write F (u) = HI (u) - //2 (u), where Hl (u) and H2 (u) are positive increasing functions such that H, (0) = a, (0) = o. Given any small positive quantity 6 we can now choose a positive number z such that 0 < Hl (u) < e, 0 < #2 (u) < e, % whenever 0 < u < z. We next write sin ku . F (u) du/u = sin ku . F (u) du/u Jo J z -f sin ku . H \ (u) du/u — sin ku . H2 (u) du/u. Jo Jo Let H (ut) denote either of the two functions Hl (u), H2 (u)] since this function is a positive increasing function the second mean value theorem for integrals may be applied and this tells us that there is a number v between 0 and z for which sin ku . H (u) du/u = H (z) \ sin ku . du/u Jo Jv I (kz — H (z) sin s . dsls . I Jkv roo r oo Since sin s . dsjs is a convergent integral, sin s . dsjs has an upper JO JT bound B which is independent of r and it is then clear that 11; sin ku . H (u) du/u < 2BH (z) < 2B€. 210 Two-dimensional Problems By the first lemma k may be chosen so large that sin ku . F (u) du/u < e, and so we have the result that re lim sin ku . F (u) du/u = 0. Ar-voo Jo It now follows that k -><X> J -a * X 4 To extend this result to the case in which the limits are — oo and oo we shall assume that for x > j3 where P1 (x) and P2 (x) are positive functions which decrease steadily to zero as x increases to oo. A similar supposition will be made for the range x < — a, the positive functions now being such that they decrease steadily to zero as x decreases to — oo. Since Pi(t) t- x' (x < /? < t < y) is a positive decreasing function of t for t> ft we may apply the second mean value theorem for integrals and this tells us that x) ^pl(p) {*sink(t_x)dt + P> - Now let \Pl(x)\< M for x > /?, then M D m ,, Pl(t)dt J-TQ - , k(p-x) -f sin By making k large enough we can make 4M/k (j8 — x) as small as we please ; moreover, this quantity is independent of y, and so we can conclude that ("sink (t-x) lim I — _ Px (t) dt — 0. k ->oo J P t •£ Similar reasoning may be applied to the integral involving P2 (t) and to the integrals arising from the range t < — a. It finally follows that foo im _>00 J-C lim or sin k (t — x) x, t — x }oo rk dt ' -oo JO ~ x)f(t)ds. Fourier's Integrals 211 To justify a change in the order of integration it will be sufficient to justify the change in the order of integration in the repeated integral I °° dt f cos s (t - x) Pl (t) ds, Jq JO where q > /?, for the other integral with limit — oo may be treated in the same way and a change in the order of integration for the remaining integral between finite limits may be justified by the standard analysis. Now let us assume that \mPi(t)dt ...... (A) Jq exists, then f "dt j k cos s(t- x) Pl (t) ds~ \K ds Tcos s(t-x) Pl (t) dt < 2k f "X (t) dt. Jq JO JO Jq Jq But, since the integral (A) exists we can choose q so large that is as small as we please. The order of integration can therefore be changed and so we have finally TT/(X) = \ds\ coss(t~ x)f(t) dt. JO J -oo The assumptions which have been made are: (1) For x > p, f(x) = Pj (x) - P2 (x), where P1 (x) and P2 (x) are positive decreasing functions integrable in the range ()8, oo). (2) A corresponding supposition for x < — a. (3) / (x) of bounded variation in a range enclosing the point x. (4) / (x) discontinuous at only a finite number of points in (— a, f3) and j / (x) | integrable in (— a, /2). § 3-13. To illustrate the method of summation we shall try to find a potential which is zero when x = 0 and when x = 1 . We shall be interested here in the case when the potential has a logarithmic singularity at the point x = x', y = 0. We first note that M (m, x') is a simple combination of primary solutions and by an extension of the method of images used in the solution of physical problems by means of primary solutions we may satisfy the boundary condition at x = 0 by means of a simple potential of type M (m, x') — M (m, — x'). This can be written in the form 2e~my sin (mx) . sin (mx')> and it is readily seen that the boundary condition at x = 1 may be satisfied by writing m = mr, where n is an integer. We now multiply by a function 14-2 212 Two-dimensional Problems of n and sum over integral values of n. To obtain a series which can be summed by means of logarithms we choose/ (n) = l/n so that our series is co 1 K = £ - e~nny [cos /ITT (x — x') — cos n-n (x -f #')] • n~-in If // > 0 the sum of this scries is* p i , cosh (rry) — cos TT (# 4- a;7) ~ COsh (TT?/) — COS TT (x ~ x') ' To extend our solution to negative values of y we write it in the form on O V — £ g~w;r|y| sjn {n7lX} sin The expression for V may be written in an alternative form which shows that it may be derived from two infinite sets of line charges arranged at regular intervals. This expression shows also that the potential V becomes infinite like — \ log [(x — x')2 -f i/'2] in the neighbourhood of x = x' ', y — 0, it thus possesses the type of singularity characteristic of a Green's function and so we may adopt the following expression for the Green's function for the region between the lines x — 0, x — 1, when the function is to be zero on these lines oo j G (x,x'\y,y') = £ - e -nrr\v-v'\ s[n (nnx) sin n - 1 n A corresponding solution of the equation is obtained by writing exp [— | y — y' \ (nV2 in place of exp [— UTT \ y — y' \] and 2n/(n* - k*/7r2) in place of the factor 2/n. § 3-14. As another illustration of the use of the simple solutions of Laplace's equation we shall consider the problem of the cooling of the fins of an air-cooled airplane engine when the fins are of the longitudinal type. The problem will be treated for simplicity as two-dimensional. A fin will be regarded as rectangular in section, of thickness 2r, and of length a. Assuming that the end x = 0 is maintained at temperature 00 by the cylinder of the engine and that it is sufficient to assume a steady state, 329 o26 the problem is to find a solution of ^~2 -f ^ = 0 and the boundary con- * See, for instance, T. Boggio, Rend, ' Lombardo (2) 42:611-624 (1909). Cooling of Fins 213 ditions * -- = 0 along y = 0, k ~— = — q0 along y = T, A y- = — <?# along # = a. The first two of these three conditions are satisfied by writing 00 0=2 Am cosh [sm(x-cm)] cos (smy), ksm r tan (sm r) - q. m-l This equation gives oo1 values of sm and when sm has been chosen the corresponding value of cm is given uniquely by the equation ksm tanh [sm (cm - a)] = y which will ensure that the third condition is satisfied. To make 0 = 0Q when x = 0 we have finally to determine the constant coefficients Am in such a way that 00 00 = S ^4W cosh (smcm) cos (sm?/). /« - 1 • This may be done with th£ aid of the orthogonal relations I cos (ysm) cos (ysn) dy = 0, m ^ n m = n. Therefore Am = 400 sech cosh (^y) sin (smr). v inj) Vm ; Harper and Brown derive from this expression a formula for the effectiveness of the fin, which they define as the ratio H/HQ) where Hn = 2q (a + r) 0n, H = a \0dS. For numerical computations it is convenient to adopt an approximate method in which the variation of 9 in the y direction is not taken into consideration. Results can then be obtained for a tapered fin. The approximate method has been used by Binnie| in his discussion of the problem for the fins of annular shape which run round a cylinder barrel. § 3-15. For some purposes it is useful to consider simple solutions of a complex type. Thus the equation dv _ d2v Si = " dx* * The formal solution is obtained by D. R. Harper and W. B. Brown (N.A.C.A. Report, No. 158, Washington, 1923), but is not used in their computations, f Phil. Mag. (7), vol. n, p. 449 (1926). 214 Two-dimensional Problems is satisfied by v = Ae"**(l+l>**, if 2i//?2 = cr. Retaining only the real part we have* v = Ae-*xco& (at - px). ...... (A) This solution is readily interpreted by considering a viscous liquid which is set in motion by the periodic motion of the plane x = 0, the quantity v being velocity in one direction parallel to this plane (§ 2-56). The pre- scribed motion of the plane x = 0 is v = A cos at = F, say. The vibrations are propagated with velocity or/j3 in the direction per- pendicular to the plane but are rapidly damped , for the amplitude diminishes in the ratio' e~2rr as the wave travels a distance of one wave-length 27r/j3. For an assigned value of a this wave-length is very small when v is very small, when v is assigned the wave-length is very small if a is very large. The equation (A) has b,een used by G. I. Taylorf-to represent the range of potential temperature at a height x in the atmosphere, the potential temperature being defined as usual, as the temperature which a mass of air would have if it were brought isentropically (i.e. without gain or loss of heat and in a reversible manner) to a standard pressure. The following examples to illustrate the use of the solution (A) are given by G. Greenf. Suppose that two different media are in contact, the boundary surface being x -= a and the boundary conditions v 9^, „ 3V2 x v>="" **£ = *•& for*=a- Let there be a periodic source of "plane -waves" on the side x, then the solution is of type Vl = 6e~^x cos (at - fax) -f AOeW*-**) cos [at + fa(x— 2a)], x < a, v2 - Bde-t*(*-c) cos [at - fa (x - c)], x > a, where c = a[l - (i/^)*], fa = (cr/2^)*, & = (<7/2i/8)*, pA = K, vS - #2 vX> PB = 2^ yV2> P = #1 V"2 + #2 vX- There is, of course, the physical difficulty that the expression for the incident waves becomes infinite when x = — oo. If we take the associated problem in which the incident waves corre- spond to a periodic supply of heat q cos at at the origin, the solution is (y2/a)* Be-**(x-c) cos (kt - fax + fac - 7r/4), where A and B have the same values as before. It is noteworthy that A * The theory is due to Stokes, Papers, vol. m, p. 1. See Lamb's Hydrodynamics, p. 586. t Proc. Eoy. Soc. London, A, vol. xciv, p. 137 (1918). t G. Green, Phil Mag. (7), vol. in, p. 784 (1927). Fluctuating Temperatures 215 and B are independent of a and that when the expressions in these solutions are integrated with respect to a from 0 to oo the physically correct solution for the case of the instantaneous generation of a quantity of heat q at the origin at time t == 0 is obtained in the form EXAMPLES 1. In the problem of the oscillating plane the viscous drag exerted by the fluid is, per unit area, . / 1 /717\ (sin at — cos at). [Rayleigh.] 2. Discuss the equations ( V -H -- ~5T \ a) =* a sin <^, (<^ a constant), where Q is a constant representing the angular velocity of the earth, and <f> is the latitude. [V. W. Ekman.] § 3-16. The solution (A of § 3-15) may be generalised by regarding A as a function of /? and then integrating with respect to /S. A solution of a very general character is thus given by v = I °V*te cos [3x - + e -^ sin [jSa; - 2j/j32*] e/r (j8) rfjS, Jo where (/> (j3) and ^ (j8) are arbitrary functions of a suitable character. Solutions of this type have been used by Rayleigh and by G. Green. Some useful identities may be obtained by comparing solutions of problems in the conduction of heat that are obtained by two different methods when the solution is known to be unique. For instance, if we use the method of simple solutions we can construct a solution 2jr t; = - S et{*-«n-^a/(f)df 7Tn»-oo Jo which is periodic in x with the period 2?!. When t = 0 the series is simply the Fourier series of the function / (x) and the inference is that with a suitable type of function / (x) our solution 216 Two-dimensional Problems is one which satisfies the initial condition v = / (x) when t — 0. Now such a solution can also be expressed by means of Laplace's integral t; - rco e M f(t)d(9 J -co and this may be written in the form 00 ft, --- 2 e 4*< JO 71= — oo When the order of integration and summation can be changed, a comparison of the two solutions indicates that , (x - £_+ gftTT,2 2 e"1**-^-"21'* = (77/1/0* S This identity, which is due to Poisson, has recently, in tho hands of Ewald, become of great importance in the mathematical theory of electro- magnetic waves in crystals. The identity can be established rigorously in several ways : (1) With the aid of Fourier series. (2) By the calculus of residues. (3) By the theory of elliptic functions (theta functions). (4) By means of the functional relation for the f -function, Riemann's method of deriving this functional relation being performed backwards. An elementary proof based on the equations x n >ex astt->oo if #„->#, ...... (1) 2~2nnH j^^-ig-acz as n -^ oo if rn~^ -> x ...... (2) \n -h r) v ; has been given recently by Polya*. We have2n~1< n\ for n= 1,2,3,... and so, forO<.r< 1, e** = 1 + ~ + ... < 1 + 2x + 2x* + ... = 1±5. 1 ! 1 — x Also, for 0< x< {, 1 4- T Sr3 7^3 ^J_ = 1 + 2x + 2x2 + ~ — < 1 + 2x 4- 2x2 + ^ Therefore e-2x-x* < _~- < e-2* * G. P61ya, Berlin, Akad. Wiss. Ber. p. 158 (1927). Frisson's Identity 217 On account of the symmetry of the binomial coefficients it is sufficient to prove (2) for r > 0. In this case . j. , 2\ / r — 1\ . j. , 2\ / r — n (l ~ n) V ~ n) '" \l ~ ~^ r/ 1W 2X / r-_l\ V n)\ n) \ n / V' the upper estimate in (B) having been applied. A use of the lower estimate gives an analogous result, which, on account of the fact that r4n~3 -> 0, com- pletes the proof of (2). Putting x = ZCDV = ze2irtv'1 in the identity k / fyvn \ we obtain S [(V(za>v) 4- 1 A/(z"")]2m = 1 % ( , ) zl¥, -1<2»<1 v--k \m + W/ where k = [m/l] is the integral part of m/l. Now let s be an arbitrary fixed complex number and t a fixed real positive number. Putting I = V[(mt)]> z = gS^» an(l dividing the series by 22m, we obtain the relation 8P~\ 2 coshM--~^n = 2 Jl + rA™l^+ I [Vim] ^ (C) Applying the limit (1) on the left and (2) on the right we finally obtain a + 2viv 00 -jj-y CO V— -00 —00 which is a form of Poisson's formula. To justify the limiting process which has just been performed in which the limit is taken for each term separately, it is sufficient to find a quantity inde- pendent of m which dominates each term in each series. There is little difficulty in finding a suitable dominating quantity for the terms on the right-hand side, but to find a suitable quantity for the terms on the left-hand side x>f the equation P61ya finds it necessary to prove the following lemma. Given two constants a and b for which a > 0, 0 < b < IT, we can find two other constants A and B such that A > 0, B > 0 and | cosh z | < eAxZ~Bv*9 when — a < x < a, — b<y<b and z = x + iy. We have, in fact, | cosh z |2 = \ (1 + cosh 2x) - sin2*/, but, for — a < x < a, we have i <1 4- cosh 2x) < 1 -f- i 2 — T^-T-r~ = 1 -f 2Ax2y say. n-i (2n)\ 218 Two-dimensional Problems On the other hand, since sin y/y decreases as y increases from 0 to TT, we have for - 6 < y < 6, ^ > ™* = Vfff, 8ay, V25>0. It follows from the inequalities that have just been established that | cosh z |2< 1 + 24z2 - 2£?/2< e2<^2-*"2), and this proves the lemma. To apply the lemma to the series (C) we note that in the first member | TTV/l | < 77/2, we therefore take b = 77/2. If s is real, the sum in the first member of (C) is dominated by the series I e-gr F-. V— —00 It is easy to dominate this series by one free from m. The case in which s is not real can also be treated in a similar manner. \ EXAMPLES 1. If P = [ °° e~x* [cos (xO) - sin (x9)] cos (2**02) d6, JO Q = / °° e-^ [cos (x^) + sin (xd)] sin (2K^2) d0, JO show by partial integration that xP=>-2KtlQ, xQ = -1Kt*£. ox ox Show also that, as t -> 0, 4P-v4Q->rr*(^ri, and that consequently 4P = 4C = 7r*(,cO~>e-a;2/4**. 2. If C - ( °° e-^v cos (y2 - a2) dt/, 7 -« /• oo ^f^ e-^ sin (y2 - a2) (^y, J -a prove that (7 + S = V(w/2), 0-^ = 2 I * e29* dd. [G. Green.] § 3-17. Conduction of heat in a moving medium. When the temperature depends on only one co-ordinate, the height above a fixed horizontal plane, and the vertical velocity of the medium is w, the equation of conduction is do 90_ d*e dt + Vdy~K~dy*' ...... (A) where K is the diffusivity. When v is constant the equation possesses a simple solution of type /i + v\ = *A2, Conduction in a Moving Medium 219 which may be generalised by summation or integration over a suitable set of values of /*. In particular, if we regard 0 as made up of periodic terms and generalise by integration over all possible periods, we obtain a solution roo 0 =, eay [f(a) sin (by -f- at) + g (a) cos (by + at)] da, .70" where 2*a = v — w, bw = — a, and / (a), g (a) are &c*itable arbitrary functions. The integral may be used" in the Stieltjes sense so that it can include the sum of a number of terms corresponding to discrete values of a. When v varies periodically in such a way that v = u (1 4- r COB at), where u, r and a are constants, a particular solution may be obtained by ...... (B) where / (t) is a function which is easily determined with the aid of the differential equation. When v is an arbitrary function of t the equation (A) has a simple solution of type which may be generalized into 0 = [ °° ewl» -'""I ~ •** F(s)ds t J — oo where F(s) is a suitable arbitrary function of s. The-solution (B) has been used by McEwen* for a comparison of the results computed from theory with the results of a series of temperature observations made off Coronado Island about 20 miles from San Diego in California. The coefficient K is to be interpreted as an "eddy conductivity" in the sense in which this term is used by G. I. Taylor. This is explained by McEwen as follows : At a depth exceeding 40 metres the direct heating of sea water by the absorption of solar radiation is less than 1 per cent, of that at the surface. Also, the temperature range at that depth would bear the same proportion to that at the surface if the variation in rate of gain of heat were due only to the variation in this rate of absorption. The direct absorption of solar radiation cannot then be the cause of the observed seasonal variation of temperature, which amounts to 5° C. at a depth of 40 metres and exceeds 1° at a depth of 100 metres. Laboratory experiments show, moreover, * Ocean Temperatures, their relation to solar radiation and oceanic circulation (University of California Semicentennial Publications, 1919). 220 Two-dimensional Problems that the ordinary process of heat conduction in still water is wholly in- adequate to produce a transfer of heat with sufficient rapidity to account for the whole phenomenon. It is now generally recognised that a much more rapid transfer of heat results from an alternating vertical circulation of the water in which, at any given instant, certain portions of the water are moving upward while others are moving downward. The resultant flow of a given column of water may be either upward or downward, or may be zero. The motion may be described as turbulent and a vivid picture of the process may be obtained by supposing that heat is conveyed from one layer to another by means of eddies. This complicated process produces a transfer of heat from level to level which, when analysed statistically, will be assumed to be governed by the same law as conduction except that the "eddy conductivity" or "JMischungsintensitat " will depend mainly on the intensity of the circulation or mixing process. An equation which is more general than (A) has been obtained by S. P. Owen* in a study of the distribution of temperature in a column of liquid flowing through a tube. Assuming, as an inference from Nettleton's experiments, that the shape of the isothermals is independent of the character of the flow, Owen con- siders an element of length 8?y fixed in space and estimates the amounts of heat entering and leaving the element across its two faces perpendicular to the y-axis to be *{-*%+'»'* and ^{-*^(0 + ^ respectively, where A is the area, p the perimeter of the cross-section of the tube, 9 the temperature of the element, E the emissivity, k, p and s the thermal conductivity, density, and specific heat of the liquid respec- tively, and where 00 is the temperature of the enclosure which surrounds the tube. Owen thus obtains the equation A k * ~ psv y 8y -EP(0- *o) % = Aps Sy, «2n-»f-?('-*o)=4' dy* dy Aps{ °' dt* where a2 - k/ps. EXAMPLES 1. Prove that a temperature 0 which satisfies the equation de de dze Ht ri~ = K 3^* ' and the conditions ^ 0 = 0 when z, = 0, 0 « 0, when y * 6, 0 . 0 when f - 0, * Proc. London Math. Soc. vol. xxm, p. 238 (1925). Wave Propagation by an Electric Cable 221 is given by the formula ttnz*z*!b*) + (i?«/4«)}M. [Somers, Proc. P%$. &>c. London, vol. xxv, p. 74 (1912); Owen, foe. ci7.] 2. If in the last example the receiver is maintained at a temperature which is a periodic function of the time, so that the condition 9 = &l when y — b is replaced by d = 6 cos a>t when y = b, the solution is B = aev(i/-&)/2* (cosh 2nb — cos 2w6)-1 [(cos m£ cosh wiy — cos my cosh TI£) cos o>£ — (sin wi| sinh 7117 — sin m^ sinh T?£) sin a>t] 4- 2e&~2 2 ( p-1 where = {(v/2K)* -f (a;/^)2} {i tan-1 (4^a./?;2)}. 7i sin § 3-18. Theory of the unloaded cable. Consider a cable in the form of a loop (Fig. 13) having an alternator A at the sending end and a receiving instrument B at the receiving end. We shall suppose that the alter nator is generating a simple periodic electromotive force which may be represented as the real part of the ° expression Eelni, where E and n are constants. Naturally, we arc interested only in the real part of any complex quantity which is used to represent a physical entity. Now, if CSx is the capacity of an element of length 8x with regard to the earth, the capacity of a length ox with regard to a similar element in the return cable must be ^Cox. Hence, if 70 is the current in the alternator and VQ the potential difference of the two sides of the cable at the sending end' lcaF0 a/0 ~*C W~~~ ~dx' Now VQ is the difference between the generated electromotive force Eeint and the drop in voltage down the alternator circuit and a capacity CQ in series with it, consequently we have the equation + F0 = Assuming that / can be expressed as the real part of X (x) eint and that / = /0 at the receiving end, we find on differentiating the last equation with respect to t and multiplying by C0 , (1 - CQL0n* + inCQRQ) 70 -f C0 -° 222 Two-dimensional Problems Hence the boundary condition at the sending end is - ox where hQC0 = 0(1- C0L0n* -f inC0R0). Similarly, if /j is the current at the receiving end and if the receiving apparatus is equivalent to an inductive resistance (L19 RJ in series with a capacity Cl , we have the boundary condition 9/1- where A^ = C (1 - C^n2 + Assuming that there is no leakage, the differential equation for /is * and if X = K1 cos /z (I — x) + K2 sin /z (I — x), where I is the distance between the alternator and receiving instrument, and Kl9 K2 are constants to be determined, we have /x2 = C (n*L - inR). Writing /x = a -f i/3, where a and /J are real, we have a2 - )82 = LCn2, 2ap = - CRn, and so, if R2 -f n*L2 = G2, we have 2«2 =Cn(G + nL), 2^82 - Cn (G - nL). When nL is large in comparison with R we may write and we have a = the wave- velocity being (CL)~^. In this case the wave-velocity and attenuation constant are approximately independent of the frequency, consequently a wave-form built up from waves of high frequency travels with very little distortion. The constants JK1 and K2 are easily determined from the boundary- conditions and we find that where F = (/^/^ — 4/I2) sin pi + 2/* (h0 + k^ cos pi. When E = 0 the differential equation possesses a finite solution only when F = 0 and this, then, is the condition for free oscillations. The roots of the equation F = 0, regarded as an equation for n, are generally complex. * Our presentation is based upon that of J. A. Fleming in his book, The propagation of electric currents in telephone and telegraph conductors. Roots of a Transcendental Equation 223 This may be seen by considering the special case when (70 = Cl = oo. This means that there are short circuits in place of the transmitting and re- ceiving apparatus. We now have A0 = Ax = 0, p? sin [d = 0, and if we satisfy this equation by writing p,l — SIT, where s is an integer, the equation sW = pip = l*C (n*L - inR) gives complex values for n. When J?0 = Rl = R the roots of the equation for n are all real. This may be proved with the aid of the following theorem due to Koshliako v * . m n Let <£0 -f iiff0 - 2 ms log (z - z8) - S k8 log (z - £s) 5-1 S-l be the complex potential of the two-dimensional flow produced by a number of sources and sinks, the sources being all above the axis of x and the sinks all on or below the axis of x. Writing zs = aB + ibs, £, = £,- iij, , where a8 , b8 , gs , r}3 are all real, we shall suppose that bs > 0, T]S > 0, ms > 0, ks > 0. Now suppose that when x is real and complex n Ir' _ y ^w*t n, r-fe -/<*> + v(»). S-l I*' — fcj * where / (x) and 0 (x) are real when x is real. If we superpose on the flow produced by the sources and sinks a rectilinear flow specified by the stream- function fa = x — y tan o>, the stream -function of the total flow is ^ = i/jQ -f fa and the points in which a stream -line iff = 6 cuts the axis of x are given by the transcendental equation or g (x) cos (x - d) + f (x) sin (x - 0) = 0. We wish to show in the first place that the roots of this equation are all real. Writing G (x] - iF (x IT (X) - lit (X = _ , . s«l ^ bs *Vs/ we have G (a:) -f- iF (x) = ez (x-d) [/(*?) + *V («)], 0 (a?) - *T (x) = e' <«-*> [/ (x) - t^ (a?)], ^ («) = / (») sin (a; ^ «) + gr (a:) cos (x - 0), G(x)=f (x) cos (a: - 6) - g (x) sin (a; - 6). * Mess, of Math. vol. LV, p. 132 (1926). Koshliakov considers only the case m, - 1, k, = 0 without any hydrodynamical interpretation of the result. 224 Two-dimensional Problems Hence, if z = x 4- iy is a root of the equation F (z) — 0, we have for this root % Now let M^ and M2 be the moduli of the -expressions on the two sides of the equation, then the equation tells us that M^ = J/22, but M * _ e- 1 ~e and from these equations it appears that y > 0, M-f < M22, while if y < 0, M !2 > J/22. Hence we must have y = 0, and so the roots of the equation ^T (2) = 0 are all real. Let us next determine the effect on the roots of varying the value of 6. If a; is a real root of the equation F (x) = 0, we have (dx/d9) [/' (x) sin (x - 9) + g' (x) cos (x - 0) -f- G (x)] - G (x), , . cos (x — 0) sin (x — 6) 1 f(*\ r^i T^T'i' therefore (<te/rf0) [/ (a?) y' (x) - f (x) g (x) + {(7 (a;)}2] - [6y (a-)]2. Now r; , -!,,-= S ( * ., - - i/'yi I Q rt I 'Y\ •.\/y ATT — *? /i i* / it*/ 1 "f~ t-j/ (».</ / ^ _a j \»</ tt'g ^^5 ^ / (a:) — ig (x) s=i\x ~ as + ^5 ^ therefore »-/»£(*) > The right-hand side is clearly positive and so dxfdO is positive for all real values of 6. This means that when x increases, the point in which the stream -line meets the axis of x moves to the right (i.e. the direction in which x increases). If we increase 6 by JTT, F (x) is transformed into — G (x), and if we add another £77- to 0, the function — G (x) is transformed into — F (x), conse- quently we surmise that the roots of F (x) = 0 are separated by those of G (x) = 0. To prove this we adopt Koshliakov's method of proof and calculate the derivative d F (x) J (x) g'J^l^f (x) g (x) + [F (x)]* + [G (*)]» dxG(x)~ "" ~[~ This is clearly positive for all values of x and infinite, perhaps, at the Koshliakov *s Theorem 225 roots of O (x) = 0. It is clear from a graph that the roots of F (x) are separated by those of 0 (x) = 0, for the curve consists of a number of branches each of which has a positive shape. In Koshliakov's case when ks — 0, ms = 1 the functions / (x) and g (x) are polynomials such that the roots of the equation / (x) -f ig (x) — 0 are of type zs = as -f- ibs, where bs > 0. The associated equation F (x) = 0 is now of a type which frequently occurs in applied mathematics. In par- ticular.if /(;«;) = &&-*», p (*) = (ft + ft) a, the roots of the equation / (x) -f ig (x) = 0 are ifa and i/32 and so we have the result that if j8x and /?2 are both positive, the roots of the equation (&L + &) x cos (x - 6) -f (&&> - #2) sin (a: - 0) = 0 ...... (B) are all real and increase with 6. The theorem may be applied to the cable equation by writing this in the form > , , 7 _ 2/^ fro ± Vl ~ M2 (\ + Al)} ^ (r« ~ where y0(70 = O, y^C^ = C, L0 == Now t- n - = (y0 + 2t> - and when the expression on the right is equated to zero, the resulting algebraic equation for p has roots of type a -t- ib, where b is positive, hence Koshliakov's theorem may be applied and the conclusion drawn that yil is real. Since in the present case /x2 = CLn2, the corresponding value of n is also real. When 0 = 0, x = wl, /?, = j82 = ZA, the equation (B) becomes identical with the equation »7 7 , 0 -,„. . 7 ,~x 2/£o> cos o>& — (a)'2 — /i^) «in o>^, (C) which occurs in the theory of the conduction of heat in a finite rod, when there is radiation at the ends, into a medium at zero temperature. The equations of this problem are in fact dv S^v di ~~ dx2' v=--f (x), for t = 0, — ~ + hv = 0 at x = 0, =- + hv = 0 at x = I, ox ox and are satisfied by v = e-K<°2t [A cos a>x -f B sin o>#] if - wB + hA - 0, a> (B cos a)l — A sin o>Z) + h(B sin toZ -f -4 cos o>Z) = 0. B 15 226 Two-dimensional Problems Eliminating A/B the equation (C) is obtained. The problem is finally solved by a summation over the roots of this equation, the root w = 0 being excluded. Equations similar to (A) occur in other branches of physics and many useful analogies may be drawn. In the theory of the transverse vibrations of a string we may suppose that the motion of each element of the string is resisted by a force proportional to its velocity*. The partial differential equation then becomes - ~ which is of the same form as (A) if K = RjL, c2 = l/LC. An equation of the same type occurs also in Rayleigh's theory of the propagation of sound in a narrow tube, taking into consideration the influence of the viscosity of the mediumf. Let X denote the total transfer of fluid across the section of the tube at the point x. The force, due to hydrostatic pressure, acting on the slice between x and x + dx, is 0 dp . 2 , d*x — 8 -if- ax = a2pdx -*—<, , dx r dx2 where 8 is the area of the cross-section, p is the pressure in the fluid, p is the density and a is the velocity of propagation of sound waves in an unlimited medium of the same material. The force due to viscosity may be inferred from the investigation for a vibrating plane (§ 3-15), provided that the thickness of the layer of air adhering to the walls of the tube be small in comparison with the diameter. Thus, if P be the perimeter of the inner section of the tube and F the velocity of the current at a distance from the walls of the tube, the tan- gential force on a slice of volume Sdx is, by the result of (§3-15, Ex. 1), equal to where n/27r is the frequency of vibration. o v- Replacing VS by -~r we can say that the equation of motion of the ut fluid for disturbances of this particular frequency is - or S , = a2 -«--, . cte2 * Rayleigh, Theory of Sound, vol. I, p. 232. t Ibid. voL n, p. 318, Air Waves in Pipes 227 This equation has been used as a basis for some interesting analogies between acoustic and electrical problems*. We shall write it in the abbre- viated form —- dt* dt ~ dx*' Rayleigh's equation has been used recently by L. F. G. Simmons and F. C. Johansen in a discussion of their experiments on the transmission of air waves through pipesf. At the end x = 0 the boundary condition is taken to be X = X0sm(nt), ...... (D) and a solution is built up from elementary solutions of type X = Aetnt±mx, where a2ra2 = — Hn2 -f iKn. Since m is complex, we write m = a 4- ip. A solution appropriate for o y- a pipe of length I with a free end (x = I) at which x - = 0 is X = A {e~ax sin (nt - px) + e-**' sin (nt - fix')} + C {e-** cos (nt - px) -f e~ax' cos (nt - px')}, where xf = 21 — x, and where the constants A, C are chosen so that X0 = A {1 + e-2*' cos 2 pi} -f Ce-™ sin 2ply 0 = - Ae-**1 sin 2pl + G {1 4- e~*al cos 2 pi} . These equations give ^r = (1 + e~™ cos 2pl) XQ, G - e-*1 sin 2pl . Z0, where T = 1 -f 2e~2aZ cos 2j8/ -f e~^1. In the case of a pipe with a fixed end the boundary condition is X = 0 at # = Z, anc we write X = A [>-«* sin (w< - px) - e-**' sin (^ - px')] -f (7 [e~aa! cos (nt - £#) - e—*' cos (n^ - j8a?')]- The boundary condition (D) is now satisfied if Z0 = A {1 - e-2*' cos 2pl} - Ce~™ cos 2j8«, 0 = Ae~™ sin 2j8Z + G {1 - e-2*' cos Therefore G^^L = (1 - e-** cos 2^) Jf09 GC7 = - XQe-** sin 2)81, where G = 1 - 2e"2ttZ cos 2^8Z -f- e^1. * See a recent discussion by W. P. Mason in the Bell System Technical Journal, vol. vi, p. 258 (1927). t Advisory Committee far Aeronautics, vol. n, p. 661 (1924-5) (E.-M. 957, Ae. 176). 15-2 228 Two-dimensional Problems If y denotes the ratio of the specific heats for air, the pressure at any point exceeds the normal pressure p0 by the quantity where X = £S. The excess pressure at the fixed end is consequently P-Po = 2&«~" YPo («2 + £»)* °~l sin (nt ~ & + #). . . , pA + aC where tan ^ = a^ T^' The following conclusion is derived from a comparison of theory with experiment : "Marked divergence between observed and calculated results shows that existing formulae relating to the transmission of sound waves through pipes cannot be successfully employed for correcting air pulsations of low frequency and finite amplitude." § 3-21. Vibration of a light string loaded at equal intervals. In recent years much work has been done on methods of approximation to solutions of partial differential equations by means of a method in which the partial differential equation is replaced initially by a partial difference equation or an equation in which both differences and differential coefficients appear. Such a method is really very old and its first use may be in the well-known problem of the light string loaded at equal intervals. This problem was discussed by Bernoulli* and later in greater detail by Lagrange|. Let the string be initially along the axis of x and let the loading masses, which we assume to be all equal, be concentrated at the points x = na, n = 0, ± 1, ± 2, ____ Let yn be the transverse displacement in a direction parallel to the «/-axis of the mass originally at the point na, then if the tension P is re- garded as constant, we have for the motion of the nth particle amyn = P (yn+l - yn) + P (yn_^ - yn). Writing k2am = P, the equation becomes tin - & (yn+1 + yn_i - 2yn). ...... (A) Let us now put uzn = yn, u2n+1 = k(yn- yn+1), then ii2n = k (u2n^ - u2n+1), or, if s is any integer, * Johann Bernoulli, Petrop. Comm. t. m, p. 13 (1728); Collected Works, vol. in, p. 198. t J. L. Lagrange, Me'canique Analytique, 1. 1, p. 390. Vibration of a Loaded String 229 This is a difference equation satisfied by the Bessel functions and a particular solution which will be found useful is given by* us = AJ8^ (2kt), where A and a are arbitrary constants and oo (__\s (l~\m+2s '•M-.?.*-.-! nff.+ D <B) Let us first consider the ideal case of an endless string and suppose that initially all the masses except one are in their proper positions on the axis of x and have no velocity, while the particle which should be at x = na has a displacement yn = ^ and a velocity yn = v, then the initial conditions are 7 7 u2n = v, u2n+l - £77, u2n_i =• - £??, while us is initially zero if s does not have one of the three values 2n — 1, 2/r, 2n -f 1. A solution which satisfies these conditions is us = v JS_2M (2fc) + fey [Js-2n-i (2fo) - e/5_2n+1 (2fa)L for, when t = 0, JT (2fe) is zero except when r = 0 and then the value is unity. When all the masses have initial velocities and displacements the solution obtained by superposition is us = XvnJs_2n (2kt) -f k^n [J^^ (2kt) - J8_2n+l (2kt)] (C) If vn = 0 we find by integration that ys = Si,n J2s_2n (2fcJ). (D) Let us now discuss the case when this series reduces to one term, namely, the one corresponding to n = 0. Referring to the known graph of the function J2s (2kt), to known theorems relating to the real zeros and to the asymptotic representation f J28 (2kt) = (7r&)-*cos(2 to - -^-j- »). (E) we obtain the following picture of the motion : The disturbed mass swings back into its stationary position, passes this and returns after reaching an extreme position for which | y | < ^ . Its motion always approaches more and more to an ordinary simple harmonic motion with frequency initially greater than &/TT, but which is very close to this value after a few oscillations. The amplitude gradually decreases, the law of decrease being eventually (rrkt)"^. This diminution * T. H. Havelock, Phil. Mag. (6), vol. xix, p. 191 (1910); E. Schrodingor, Ann. d. Phys. Ed. XLIV, S. 916 (19H); M. Koppo, Pr. (No. 96) Andreas- Realgymn. Berlin (1899), reviewed in Fortschritte der Math. (1899). f Whittaker and Watson, Modern Analysis, p. 368. The formula is due to Poisson. An ex- tension of the formula is obtained and used by Koppe in his investigation. The complete asymptotic expansion is given in Modern A nalysis. 230 , Two-dimensional Problems depends on the fact that the vibrational energy of the mass is gradually transferred to its neighbours, which part with it gradually themselves and so on along the string in both directions. After a long time, when 2kt is so large that the asymptotic representation (E) can be used for the Bessel functions of low order, the masses in the neighbourhood of the origin vibrate approximately in the manner specified by the "limiting vibration" of our arrangement, neighbouring points being in opposite phase. The amplitudes, however, decrease according to the law mentioned above and the range over which this approximate description of the vibration is valid gets larger and larger. According to the formula (D) all the masses are set in motion at the outset, and all, except the one originally displaced, begin to move in the positive direction if T?O > 0. Let us consider the way in which the mass originally at x = na begins its motion. The larger n is, the slower is the beginning of the motion and the longer does it continue in one direction. This is because J2n (2kt) vanishes like Ant2H, as / approaches zero, An being the constant multiplier in the expansion (B). Also because the first value oi 2kt for which the func- tion vanishes lies between ^/{(2ri) (2n -f 2)} and V{(2) (%n + *) (2^ + 3)}. It is interesting to note that in this elementary disturbance there is no question of a propagation with a definite velocity c as we might expect from the analogous case of the stretched string. Let us, however, examine the case in which all the particles are set in motion initially and in such a way that the resulting motion is periodic. Writing yn = Yne2lku>t we have the difference equation If a) = sin <f> this equation is satisfied by Yn = A sin 2n<f> + B cos 2n<f>. ...... (F) Choosing the particular solution Y = Ce~2in* •*• n — ^c 9 we have yn = Ce2i(ktB{n*~n*}. Making kt sin <£ — n<j> constant we see that the phase velocity is _ ak sin <f> «-—?—• The period T is given by the equation T = — __ k sin </> ' and the wave-length A by the equation A = cT « ira/<f>. Group Velocity 231 The phase velocity thus depends on the wave-length and so there is a phenomenon analogous to dispersion. Introducing the idea of a group velocity U such that ~x ^, 3A r/9A 3* + Udx~^ that is, such that A does not vary in the neighbourhood of a geometrical point travelling with velocity U, we next consider a geometrical point which travels with the waves. For this point A varies in a manner given by the equation * ^ ~^ ~ , a ^ ^A _ x dc _ , dc cc a7 + Cfo fo 5Aai' the second member expressing the rate at which two consecutive wave- crests are separating from one another. Eliminating the derivatives of A we obtain the formula of Stokes and Rayleigh, Z7 = c-A~=tf(A),say: In the present case U = ak cos (7ra/A) = ak cos <f>. When A-> oo, U -> ak = U(co) = c(oo). Hence for long waves the group velocity is approximately the same as the wave velocity. For the shortest waves <f> = \TT, we have U = 0. When there are only n masses the two extreme ones being at distance a from a fixed end of the string, the equations of motion are & + & (2ft - 0 - ft) = 0, $2 + & (2y2 - ft - ft) = 0, Assuming ys = Y8e2ik<at as before and eliminating the quantities Y8 from the resulting equations we obtain the following condition for free oscillations : 2 cos 2<f> - 1 0 0 | = 0, - 1 2 cos 2^ — 1 0 0 - 1 2 cos 2<j> - 1 t where there are n rows and columns in the determinant. Since it is readily shown by induction that n ~ sin 2<f> * See Lamb's Hydrodynamics, p. 359. 232 Two-dimensional Problems This is zero if 2 (n -f- 1) (/> = rvr, r = 1, 2, ... n; we thus obtain /i different natural frequencies of vibration. When the motion corresponding to any one of these natural frequencies is desired we use an expression of type (F) for Yn and the end condition Yn = 0 will be satisfied by writing B = 0. Hence one of the natural vibrations is given by ym = A sin2w<£e2i*<8ln*, where 2 (n + 1) (f> = TTT (r = 1, 2, ... n). If the velocity is initially zero we write ys = ^4 sin 2s<^ . cos (2&£ sin <f>). Let us examine more fully the case in which n = 2. The possible values of (f> are and -- , consequently in the first case A sin ^ cos (2Kt sin ^ j , y2 = A sin ~ cos ( 2kt sin ~ j ; yl and j/2 have the same sign and the string does not cross the axis of x. In the second case A . 2?! /rt/J . 77\ ,.477 /rt/J . 77 yl = A sm cos ( 2tct sin - , s/2 = A sin — cos ( 2kt sm - o \ o/ o \ o ^ and y2 have opposite signs, the string crosses the axis of z at its middle point which is a node of the vibration. When n = 3 we find in a similar way that there is one vibration without a node, one with a node and one with two nodes. The extension to the case in which n has any integral value is clear. The general vibration, moreover, is built up by superposition from the elementary vibrations which have respectively 0, 1, 2, ... n — 1 nodes, the nodes of one elementary vibration such as the 5th being separated by those of the (s - l)th. If we regard this solution as valid for all integral values of s, we may apply it to the infinite string. The initial value of ya is now A sin 2s<f> and so by applying the general formula we are led to the surmise that there is a relation 00 sin 2s<f> . cos (2kt sin cf>) == S sin 2p<f> . J2s-2» (2&0> J>= —00 which is true for all real values of <f>. This relation is easily proved with the aid of well-known formulae. An equation similar to (A) occurs in the theory of the vibrations of a row of similar simple pendulums (a, m) whose bobs are in a horizontal line and equally spaced, consecutive bobs being connected by springs as shown in Fig. 14. Using yn to denote the horizontal deflection of the nth Electrical Filter 233 bob along the line of bobs and supposing that the constants of the springs are all equal, the equations of motion are of type / / x 7 / x my //-.x tYlij == K (I/ I/ } rC ( I/ 11 ) — I/ (\JTI The periodic solutions of this equa- tion give a good illustration of the filter properties of chains of electric circuits that were discovered by G. A. Camp- bell*. The mechanical system may, in fact, be regarded as an analogue of the Flg> 14' following electrical system consisting of a chain of electrical circuits each of which contains elements with . inductance and capacitance (Fig. I ' . . 15). _- v/n-i — r— (*n — T— V^TIM The following discussion is — ' ' ' based largely upon that of T. B. Fig' 15' Brown f. When the chain is of infinite length and the motion is periodic $n appropriate solution is obtained by writing yn = Ar~n sin (pt — n<f>). If mg — ap2 = 2akQ the equations for r and (f> are (1 - r2) sin <f> = 0, (r2 + 1) cos <f> = 2r (1 + Q). These are satisfied by <f> = 0, r ^ 1 and by r = 1, cos <f> = 1 + Q. In the latter case there is transmission without attenuation but with a change of phase from section to section, the phase velocity corresponding to a fre- quency / = p/2n being v^px = Zirfx (h (b where x is the length of each section. This type of transmission is possible only when Q lies between 0 and - 2, that is when/ lies between /t and/2, where , . I/AJ \ / / fl\ I 4/c Q \ O— f I i & 1 9^-f / I _L y I ZTT/I = A / I - I , ^^[/2 ~~ A / I ' " I • J1 V \aJ V \w a/ This range of frequencies gives a pass band or transmission band. On the other hand, when <£ = 0 we have r = 1 -f Q ± [(1 + Q)2 - 1]*, and it is clear that r > 0 when /< fl9 r < 0 when /> /2. The negative value of r indicates that adjacent sections are moving in opposite directions with amplitudes decreasing from section to section as we proceed in one direction down the line. We may use a positive value of r if we take <f> = TT instead of <f> = 0. It should be noticed that r is real only when / lies outside the pass * U.S. Patent No. 1,227,113 (1917); Bell System Tech. Journ. p. 1 (Nov. 1922). t Spurn. Opt. Soc. America, vol. viu, p. 343 (1924). 234 Two-dimensional Problems band. There are two regions in which / may lie and these are called stop bands or suppression bands ; one of these is direct and the other reverse. The stopping efficiency of each section is represented by loge | r \ and this is plotted against /in Brown's diagram. For a further discussion of wave-filters reference may be made to papers by Zobel, Wheeler and Murnaghan. Let us now write equation (G) in the form (D2 + c2) yn - &2 (yM + yn-1)f where b*=k, c2 = ~+?, D=~4, m ma at and let us seek a solution which satisfies the initial conditions 2/0=1, * 2/o = 0* ?/i One way of finding the desired solution is to expand yn in ascending powers of 62. Writing yn = (n, n) b*n + (n,n + 2) it is found by substitution in the equation that 262 \n (/&_+_!) (n + 2) /_J 262 ~ ~ 1 ["/ 262 \n 2/71 ~~ 2» [U>2 + c2 / _ 2 . 4(2n + 2)T2rT+ 4) the law of the coefficients being easily verified. The meaning which must be given to 2 is one in which the Taylor expansion of the expression in powers of t starts with t2m . (2b2)m/(2m) I An expression which seems to be suitable for our purpose is obtained by writing cos ct = 75— ezt ~*- where C is a circle with its centre at the origin and with a radius greater than c. The result of the operation is then 2b2 m I zdz and we obtain the formal expansion - I _1_ f *t _ ldz_ [7 262 V 4. (»+1)(ra+2) /_ 2^a_V^ , I y" 2" ' 2wi Jce z2 + c2 LV32 + cV 2 (2» + 2) Ua + cV ;' + '"J ' Torsional Vibrations of a Shaft 235 In particular, 1 r eztzdz ~ eztzdz 1 r 2/1 = 2*ri ]e [>2c~ 464}* z2 + c2 -f [(z2+c2)2- and generally 1 r " ^ 2m j 2fe2 ! c [(z2 + c2)2 - 464]i (z2 + c2 + [(z2 + c2)2 - 46*]*) * It is easy to verify that this expression satisfies equation (G) and the prescribed initial conditions. When c2 = 26 2 the formula reduces to !_ f e" _ \ b ___ which must be equivalent to J2n (2bt). The solution of the equation (D + c2) yw = 62 (^^ + y,^), which satisfies the initial conditions 2/o = 1, 2/i = 0, t/2 = 0, ... when t = 0, is given by the formula = ^ f _e?*zdz An equation which is slightly more general than (A) occurs in the theory of the torsional vibrations of a shaft with several rotating masses*. This theory can be regarded as an extension of that of § 1-54 and as a preliminary study leading up to the more general case of a shaft whose sectional pro- perties vary longitudinally in an arbitrary manner. Let /!, /2, /3, ... /jv be the moments of inertia of the rotating masses about the axis of the shaft, 019 02, 03, . . . 0N the angles of rotation of these masses during vibration, kl9 k2, k3, ... kN the spring constants of the shaft for the successive intervals between the rotating masses. Then *i (*i ~ 02), ^2 (<?2 - 9*), -. *W (Vi - ON) are torque moments for these intervals. Neglecting the moments of inertia of these intervening portions of the shaft in comparison with /x , 72 , ... /# the kinetic energy T and the potential energy V of the vibrating system are given by the equations 2T=/1d1*+/A"+.../A>, 2F = k, (0, - 02)* + kt (02 - 03)2 + - **-i (0.v-i - »*)', * See S. Timoshenko, Vibration Problems in Engineering, p. 138; J. Morris, The Strength of Shafts in Vibration, ch. x (Crosby Lockwood, London, 1929). 236 Two-dimensional Problems and Lagrange's equations give /A + kn (0n - 0n+1) - kn_, (0^ - 0n) = 0, Except for the two end equations, for which n = I, N, respectively, these equations are the same as those of a light string loaded at unequal 'Intervals. When k^ ----- Jc2 = &3 = ... = kN the foregoing analysis can be used with slight modifications. A second case of some mathematical interest arises when A^ = Jfc3 = fe6 = ... fc 7. __ 7* __ 7^ __ 7, ^ 2 — 4 ~~ 6 ~~ ••• '^* EXAMPLES. 1. By considering special solutions of the equation of the loaded string prove that the following relations are indicated: 00 (n - x? = S 7/1= -CO 2. Prove that the equation J M ^ = J [^_, (x) + ^n+1 (x) - 2^n (a:)] is satisfied by Fn (x) = e^t*^) 2 7n_m (* - a) Fm (a), m^-~<x> where /n (a-) = *~Vn (ix), and obtain the solution in the form of a contour integral. 3. Prove that the equation lln = V [yn+z -H 2/n_2 is satisfied by 1 ( _ e ?w/n " 2^J Jc [(Za~+ c2)2 - 46*1 z2 + c-f [(z2 + c2)2- 4. Each mass in a system is connected with its immediate neighbours on the two sides by elastic rods capable of bending but without inertia. Assuming that the potential energy of bending is F = . prove that the oscillations of the system are given by an equation of type when r > 1 and obtain the two end equations. [Lord Rayleigh, Phil. Mag. (5), vol. XLTV, p. 356 (1897); Scientific Papers, vol. rv, p. 342.] 6. Prove that a solution of the last equation is given by Vr = Jr Q C08 (Y "" !•) * [Havelock.] §3-31. Potential function with assigned values on a circle. Let the origin and scale of measurement be chosen so that the circle is the unit Potential with Assigned Boundary Values 237 circle | z \ = 1 and let z' = eie/ be the complex number for a point P' on this circle. Our problem is to find a potential function V which satisfies the condition T7 as z~* z - To make the problem more precise the way in which the point z approaches z' ought to be specified and something must be said about the restrictions, if any, which must be laid on the function/ (0'). These points will be considered later; for the present it will be supposed simply that / (0') is real and uniquely defined for each value of 0' when 9' is a real angle between — TT and rr. The mode of approach which will be considered now is one in which z moves towards z' along a radius of the unit circle. In other words, if z = re!*, where r and 9 are real, we shall suppose that 9 remains equal to 9' and that r -> 1. Now let z = r . e~id be the complex quantity conjugate to z; an attempt will be made first of all to represent V by means of a finite or infinite series F0 = c0 + 2 (cnzn + c^i"), ...... (A) 71-1 where c0 is a real constant and cn9 c_n are conjugate complex constants. When the series contains only a finite number of terms it evidently represents a potential function and in the limiting process z-+z' it tends to the value Ecnz'n, where the summation extends over all integral values of n for which cn ^ 0. Negative indices are included because zn -> z'~n. Supposing now that the finite series represents the function/ (#'), the coefficient cn is evidently given by the formula for the integral of eim6' between — -n and TT is zero unless m = 0, conse- quently the term cnz'n in the series for/ (#') is the only one which contributes to the value of the integral. A function/ (0') which can be represented as the sum of a finite number of terms of type cne~ine' is evidently of a special nature and the natural thing to do is to endeavour to extend the solution which has just been found by considering the case in which an infinite number of the constants cn defined by the formula (B) are different from zero. The series (A) formed from these constants then contains an infinite number of terms. Let us now assume that the f unction /(#') is integrable in the interval — TT < 6' < IT. Since the series 1 + S rn [e*»<*-*'> + e-in<'-6'>] == K (9 - 9') i is uniformly convergent for all points of this interval if | r | < 1, it may be 238 Two-dimensional Problems integrated term by term after it has been multiplied by/ (0'). The potential function V may, consequently, be expressed in the form d0'9 (C) where K (co) =1 + 2 £ rn cos na> = - — ~ — --- A. n-i 1 - 2r coso> -f r2 The integral representing our potential function F is generally called Poisson's integral and will be denoted here by the symbol P (r, 6) to indicate that it depends on both r and 6. This integral is of great importance in the theory of Fourier series as well as in the theory of potential functions. The formula (C) may be obtained in another way by using the Green's function for the circle. If P (r, 6) is the pole of the Green's function, Q (r-1, 6) the inverse point and P' (/, 6') an arbitrary point which is inside the circle (or on the circle) when P is inside the circle and outside (or on) the circle when P is outside the circle, an appropriate expression for the Green's function is where A is an arbitrary point on the circle. This expression is evidently zero when P' is on the circle, it becomes infinite in the desired manner when P' approaches P and it is evidently a potential function which is regular except at P. The formula " 2~7r J o F- 2r cos~(0 ~(F)+~r* represents a potential function which takes the value / (0) on the circle and is regular both inside and outside the circle. When r = 0 the formula gives the relation where F0 is the value of F at the centre of the circle. This is the two- dimensional form of Gauss's mean value theorem. When / (6) is real for real values of 6 the formula (0) may be written in the form of the sum of two conjugate complex quantities each of which takes the value J/ (6) at the point z = eie on the unit circle, and we deduce Schwarz's more general expression ...... (D) Potential with Assigned Boundary Values 239 for a function F (z) whose real part on the unit circle is/ (6). The imaginary part of F (z) is a potential function i W which is given by the formula w - h 4- 1 (**rsm(e-6')f(e') M' 0 TrJo l-2reos(0-0') + r2' If in the formula (D) we have / (2-n — 6) = / (0) we obtain Boggio's formula for a function F (z) whose real part takes an assigned value / (0) on the semicircle z = eie, 0 < 9 < TT, and whose imaginary part vanishes on the line z = cos a, 0 < a < TT, i.e. the diameter of the semicircle, 1- 2zcos0'-f z2* When F = / (z), where / (z) is a function which is analytic in the unit circle | z' — z \ — 1, Gauss's mean value theorem may be written in the form and is then a particular case of Cauchy's integral theorem. By means of the substitution z' = pz, F(pz)--=f(z) the theorem may be extended to a circle of radius p. If on the circle we have | / (zf), \ < M, the formula shows that | / (z) | < M. More generally we can say that if / (z) is a function which is regular and analytic in a closed region G and is free from zeros in G, then the greatest value of | / (z) \ is attained at some point of the boundary of G and the least value of | / (z) \ is also attained at some point on the boundary of G. In this statement values of / (z) for points outside G are not taken into consideration at all. If/ (z) is constant the theorem is trivial. If/ (z) is not constant and has its greatest value M at some point z0 inside G we can find a small neigh- bourhood of z0 entirely within G for which | / (z) \ < M , and if O is a small circle in this neighbourhood and with z0 as centre this inequality holds for each point of C and so which leads to a contradiction. The theorem relating to the minimum value of | /(z) | may be derived from the foregoing by considering the analytic function l//(z). § 3-32. Elementary treatment of Poissorfs integral*. To find the limit lim P (ry 0) r->l it will be assumed in the first place that / (9) is integrable according to * This treatment is based upon that given in Carslaw's Fourier Series and Integrals. 240 Two-dimensional Problems Ricmann's definition and that if it is not bounded it is of such a nature that the integral f(9')dff j —it is absolutely convergent. Let us now suppose that 0 is a point of the interval — TT < 6 < TT which does not coincide with one of the end points. We shall suppose further that the limit ,. r « ,n , x «/n hm [f(6+ r) + /(0- T->0 /T^ ...... (E) exists and is equal to 2F (6), where F (9) is simply a symbol for a quantity which is defined by this limit when 6 is chosen in advance. No knowledge of the properties of the function F (9) will be required. Now let a function <D (#') be defined for all values of 6' in the interval (— TT < 9' < TT) by the equation <i> (0') = f (0') - F (0). Then (0 - 9')[f(9') - F (9)] d9' P (r, 9) - F (9) = Since, by hypothesis, the limit (E) exists, a positive number 77 can be found so as to satisfy the conditions \f(0 + a)+f(0-a)-2F(0)\<€, when 0 < a < 17, 9 — 77 > — 77, 9 + 77 < 77, e being an arbitrary small positive quantity chosen in advance. Then fo + y fi K (0 - 0') O (6') dO' = K (a) [O (0 + a) + «» (0 - a)] fl-7) JO da and so (9)] da, K(0-0')Q> (O1) ri TT < € \ K (a) da < e\ K (a) da = JO J-TT Also, when 0 < r < 1, [' ~*/c (0 - 9') O (fl') dfl' + f * ic (ff - 0') 0 (0') dfl7 -rr h + n 277 (77), say, where A is a positive quantity. But, when 0< r < 1, (1 -f 4r sin2 77/2 2r sin2 ij/2 ' Discussion of Poissorfs Integral 241 Hence 2irAK (T?) < 2-Tre if r is so chosen that Lz r__ 5 2rsin2"7p<2' and this inequality is satisfied if r> s • Combining the two results we find that \P(r,B)-F (9) | < 2e, if !>,>[, + * sin.*]'1. Hence when the limit (E) exists, P(r, 0) -*jF(0) asr -> 1. When 0 is a point of continuity of the function/ (0) we have, of course, F (0) = / (0) and so V tends to the assigned value. To prove that V is a potential function when r < 1 it is sufficient to remark that the series obtained by differentiating (A) term by term with respect to z is uniformly 0 Y convergent for r<s< 1, where s is independent of r and 0, hence -g-- 12 T/ exists and is a function of z only. The equation x— ~- = 0 then follows immediately. The behaviour of Poisson's integral in the neighbourhood of a point on the circle at which / (0) is discontinuous is quite interesting. Let us suppose that / (0) has different values f± (0) and /2 (0) when the point 0 is approached along the circle from different sides, then if the point 9 is approached along a chord in a direction making an angle air with the direction of the curve for which 0 increases the definite integral tends to the value , - /m f /m (1 -«)A(0) + «/2(0). A proof of this theorem is given by W. Gross, Zeits. f. Math. Bd. n, S. 273 (1918). When/ (0) is continuous round the circle we have the result that V -> / (0) as any point on the circle is approached along an arbitrary chord through the point. This theorem has also been proved by P. Pain- leve, Comptes Rendus, t. cxn, p. 653 (1891) and by L. Lichtenstein, Journ. f. Math. Bd. CXL, S. 100 (1911). EXAMPLES 1 Show by means of Poisson's formula that if - 1 (0 < 6 < TT), 16 242 Two-dimensional Problems the potential V is given by the equation F = 1 + 2 tan-1 ~ — .— . (r2 < a2). TT 2ar sin 0 ' 2. Let the unit circle z = e*d be divided into n arcs by points of division Bt , 02 , . . . 0n , where 0 < 0X < 62 < ... < 0n = 2*. Let <f> -f i^ = / (z) be analytic for | z | < 1 and let <£ satisfy the following conditions on the circle * = <W 0m_l<6<0mt c^-c^ cm being an arbitrary constant, then 27T/ (zj - - 27rCl 4- 2 (cm - c,) [0, + 2* log (e1^ - z)]. [H. Villat, £W/. rfe la Soc. Math, de 'France, t. xxxix, p. 443 (1911); "Aper9us the"oriques sur la resistance des fluides," Scientia (1920).] § 3-33. Fourier series which are conjugate. When r is put equal to 1 in the series (A) the resulting series may be written in the form n— —oo and is the " Fourier series" associated with the function/ (8). Separating the real and imaginary parts, the series may be written in the form ^ a0 -f S (av cos v9 + bv sin vd), ...... (A') where °* = J (0>) d0' > av = - [n f(6')coav0'd6', TT J -IT 6F= L [ T J The constants a,, 6,, are called the " Fourier constants" associated with the function/ (#'). In terms of these constants the series for V is 00 x 7 = a0-f S rv (a,, cos v0 + 6^ sin i/0). ...... (B') v-l When all the coefficients are real the series for the conjugate potential W is 60 -f- r (a± sin 0 - ^ cos 0) + r2 (a2 sin 0 - 62 cos 0) + ... , ...... (C') and this is associated with the series 60 -f (ax sin 0 — &! cos 0) + (a2 sin 0 — b2 cos 0) -f ..., ...... (D') which, when 60 = 0, is called the conjugate* of the Fourier series (A'). There is now a considerable amount of knowledge relating to the con- jugate series. One question of importance in potential theory is that of the * Sometimes it is this series with the sign changed which is called the conjugate series. See L. Fejer, Crelle, vol. CXLH, p. 165 (1913); G. H. Hardy and J. E. Littlewood, Proc. London Math. Soc. (2), vol. xxiv, p. 211 (1926). Conjugate Fourier Series 243 existence of a function g (9) of which the foregoing series is the Fourier series. In this connection we may mention a theorem, due to Fatou *, which states that if/ (6) is everywhere continuous and the potential W is expressed in the form W — W (r, 0), the necessary and sufficient condition for the existence of the limit lim W(r,0) = g(0) ...... (E') r-+l for any assigned value of 6 is that the limit lim f * [/ (0 +T) -/ (8 -r)] cot ldr = - 2^(0) ...... (F) «->OJe ^ should exist. Fatou has also shown that if / (6) has a finite lower bound and is such that / (0) is integrable in the sense of Lebesgue then the limit (F') exists almost everywhere. Lichtensteinf has recently added to this theorem by showing that the integral | J — 7 exists when ft/ J —IT exists. For further properties of Poisson's integral and conjugate Fourier series reference may be made to the book of G. C. Evans on the logarithmic potential J and to Fichtenholz's paper in Fundamenta Mathematicae (1929). Fatou's expression for g (9), when/ (6) is given, is 9 (6) = 277 In the last integral the symbol P denotes that the integral has its principal value. Villat has deduced this expression by a limiting process with the aid of the result of Ex. 2, § 3-32. The formula is quite useful in the hydro- dynamical theory of thin aerofoils. An alternative expression, obtained by an integration by parts, is g (e] = L f_/' (f) log sin2 § 3-34. AbeVs theorem for power series. When for any fixed value of 0 the Fourier series converges to a sum which may be denoted for the moment by g (6), it may be shown with the aid of a property of power series discovered by N. H. Abel that V ->• g (6) as r -*• 1. But since * Acta Math. vol. xxx, p. 335 (1906). t Crette's Journ. vol. oxu, p. 12 (1912). J Amer. Math. Soc. Colloquium Publication*, vol. vi (1927). 16-2 244 Two-dimensional Problems V -> F (0) we must have g (9) = F (0) and so the Fourier series represents F (9) whenever it is convergent. The series for V may be written in the form V = UQ + rul + r2uz+ ..., (G') and it should be noted that the coefficients r, r*, ... occurring in the different terms are all positive and form a decreasing sequence. The theorem to be proved is applicable to the more general series V = t>o^o + fli^i + v2u2 + where the factors v0, vl9 v2 are all positive and such that «Vu< ^n, v0= !• Let us write «*0 = ^0> «1 = ^0 + %> S2 = U0 + % + U2, and suppose that the quantities s0 , ^ , s2 , ... possess an upper limit H and a lower limit A, then A % sn < //, for n = 0, 1, 2, .... Now if Vn denotes the sum of the first n + \ terms of the series V, V - v0Uo H- ViUi + ... ?;w?/n = Oo - '^l) ^0 + (Vl - V2) «! + ... (Vn_! ~ Vn) 5n_! + VnSn, and in this series not one of the partial sums sm has a negative coefficient. Hence Vn < K - vi) 7/ -f (?;i ~ va) 7/ -H ••• K-i - ^n) ^ + vnH> and Kn > (?'0 - Vj) li + (i\ - v2) h + ... (vn-1 - vw) A + vwA. Summing the two series we obtain the inequality v0A -r Fn < v0//, which shows that | Vn \ < vnk, where h is a fixed quantity greater than either | h \ or | H \ . Similarly, if hnm — VmUm + vm+lum+l ~f~ ••• vm+num+n> we have the inequality I 7? m I *f 11 Is I •"'n I <• vmKm> where km is a positive quantity greater than any one of the quantities I um | , | nm -f um+l | , | um + um+1 + ^m+2 | , .... If now the series UQ 4- u^ 4- w2 + • • • ^s convergent and « is any arbitrarily chosen small positive quantity, a number m (e) can be found such that | Um + WWH-I+ ... Um+n | < €, for n ^ 0, 1, 2, ... and m > w (f). When m is chosen in this way we may take km = € and since vm < v0 < 1 we have the inequality I Rn™ I < €. AbeVs Convergence Theorem 245 When the quantity vn is a function of a variable r which lies in the unit interval 0 < r < 1 the foregoing inequality shows that the series (G') is uniformly convergent for all values of r in this interval and so represents a continuous function of r. In the case under consideration we have vn = rn and the conditions imposed on vn are satisfied if 0 < r < 1. The function V is consequently continuous at r — 1 and so UQ -f u^ + u2 + ... = lim P (r, 0) = F (9). i §3-41. The analytical character of a regular logarithmic potential*. Poisson's integral may be used to prove that a logarithmic potential V which is regular in a region D is an analytic function of x and y. We may, without loss of generality, take the origin at an arbitrary point within D. Let C denote the circle x2 -f y2 — a2 which lies entirely within D, then for a point x = a cos a, y — a sin a on this circle, V = / (a), where / (a) is a continuous function of a and so by Poisson's formula V " J0 a2 f r2- 2ar cos (0 - a) ' where x = r cos 9, y = r sin 9 and r < a. Now the series *.2 oo / r\ n ,— ,= 1 + 2 S (') cosn(0-«) a2 _|_ r2 __ 2ar cos (9 — a) tl^i \a) is absolutely and uniformly convergent and so can be integrated term by term after being multiplied by / (a) rfa/2?r. Therefore F = a0 -f 2 f- ) (an cos TI^ + bn sin ?i0). n-l \a/ Now if in the polynomial T\ - ) (an cos ?i^ + 6n sin n9) a/ = la~n K (» + *y)n + an (# - *V)W - *n (^ + %)" "I" ^n (# ~ ^)W] we replace each term of type c,vqx*'tfl ^y ^s modulus, the resulting expres- sion will be less than the corresponding expression obtained by doing the same thing to each term of type ePQxpyq in the expansion of each of the four binomials and adding the results. Now this last expression is less than (T\ - ) a/ where M is the upper bound of an and bn . Now let | x \ < ,9, | y | < s, where s < a/2, then 2 [| x | + | y \] Ma~n < 2 (2s/a)n M, and the series of moduli is convergent. The series for F is thus a power * E. Picard, Cours d* Analyse, t. n, p. 18. 246 Two-dimensional Problems series in x and y which is absolutely convergent for | x \ < s, \ y \ < s, it thus represents an analytic function. Since, moreover, the origin was chosen at an arbitrary point in D it follows that V is analytic at each point within D. For the parabolic equation ~- = y^ there is a theorem given by Holmgren * which indicates that z is an analytic function of x in the neigh- bourhood of a point (x0, yQ) in a region R within which z is regular. If through the point (x0, yQ) there is a segment a < y < b of the line x = XQ which lies entirely within J?, there is a number c such that for | x — XQ | < c, a < y < b there is an expansion //r _ T ^2n (rf __ z (x, y) = S eLL #<•> (y) + 2 where $ (y) = * (*o> y), $(y) = -faz (xo> V)- These functions <£ (?/), 0 (y) are continuous (D, oo) in a < y < 6 and their derivatives satisfy inequalities of type | </>(n) (y) | < Mc~2n (2n) !, | 0(n) (y) | < Mc~*n (2n) ! . § 3-42. Harnack's theorem^. Let JF a , for each positive integral value of ,9, be a potential function which is continuous (D, 2) (i.e. regular) in a closed region R and let the infinite series ^i 4- w2 + w3 + ... ...... (A) converge uniformly on the boundary J5 of R, then the series converges uniformly throughout R and represents a potential function which is regular and analytic in R. The sum wn -f ivn+l + ... wn+v ig a potential func- tion regular in R. If it is not a constant it assumes its extreme value on B and if Np is the numerically greatest of these we shall have | wn + wn+l+ ... wn+P I < \N9\. Since, however, the series converges uniformly on B we can choose a num- ber m (e) such that when n > m (c) we have \N,\<*. for all positive integral values of p. This inequality, combined with the previous one, proves that the series (A) converges uniformly in R and so represents a continuous function w. Now let C be any circle which lies entirely within 7? and let Poisson's formula be used to obtain expressions for potential functions W, Wly W2, W3, ... regular within C and having respectively the same boundary values as the functions W9wl9w2,w3, ... * E. Holmgren, ArJnv for Mat., Astr. och Fyaik, Bd. I (1904); Bd. m (1906); Bd. iv (1907); Comptes Rendus, t. CXLV, p. 1401 (1907). t Kellogg calls this Harnack's first theorem. See Potential Theory, p. 248. The theorem was given by Harnack in his book. It has been extended to other equations of elliptic type by L. Lichtenstein, Crelle'a Journal, Bd. CXLII, S. 1 (1913). Analytical Character of Potentials 247 Since a potential function with assigned boundary values on C is unique if it is required also to be regular within C we have Ws — ws (s = 1, 2, ...). Furthermore, since the series (A) is uniformly convergent it may be inte- grated term by term after multiplication by the appropriate Poisson factor. Therefore at any point within £ w = w,+ w2+ w3+ ... = U\ ~\~ W2 -f M>3 + ... = W. Hence within C the function w is identical with the regular potential func- tion which has the same values as w at points on C. Since C is an arbitrary circle within R it follows that w is a regular potential function at all points of R and is consequently analytic at each point of R. For recent work relating to the analytical character of the solutions of elliptic partial differential equations reference may be made to L. Lichten- steiri, Enzyklo'padie der Math. Wiss., n C. 12 ; T. Rado, Math. Zeits. Bd. xxv, S. 514 (1926); S. Bernstein, ibid. Bd. xxvm, S. 330 (1928); H. Lewy, Gott. Nachr. (1927), Math. Ann. Bd. ci, S. 609 (1929). § 3-51. Scliwarz's alternating process. H. A. Schwarz* has used an alter- nating process, somewhat similar to that used by R. Murphy f in the treat- ment of the electrical problem of two conducting spheres, to solve the first boundary problem of potential theory for the case of a region bounded by a contour made up of a finite number of analytic arcs meeting at angles different from zero. To indicate the process we consider the simple case of two contours aa, 6/3 bounding two areas A, B which have a common part C bounded by a and /?, while a and 6 bound a region D represented by A f B — C. We shall use the symbols a, by a, /j to denote also the parameters by means of which the points on these curves may be expressed in a uniform continuous manner and shall use the symbols m and n to denote the points common to the curves a and b. We shall suppose, moreover, that the choice of parameters is made in such a way that in and n are represented by the parameters m and n whether they are regarded as points on a, 6, a or /?. This can always be done by subjecting parameters chosen for each curve to suitable linear transformations. Our problem now is to find a potential function V which is regular within D and which satisfies the boundary conditions V — f (a) on a, V - g (b) on 6, where / (m) = g (m), f (n) = g (n). • We shall suppose that / (a) is continuous on a and that g (b) is con- tinuous on b. We shall suppose also that a function h (a), which is con- tinuous on a, is chosen so as to satisfy the conditions h(m) =/(m), h (n) =/(/&). * Berlin MonatsberichU (1870); Gesammelte Werke, Bd. n, S. m t Electricity, p. 93, Cambridge (1833). 248 Two-dimensional Problems We now form a sequence of logarithmic potentials u1}u2) ... regular in A, and a sequence of logarithmic potentials vl9 v2, ... regular in J3; these potentials being chosen so as to satisfy the following boundary conditions in which us (/?) denotes the value of us on /3, and vs (a) denotes the value of vs on a, (s ~ 1, 2, ...): on a, u^ — ti (a) on a, u2 = / (&) on a, i£2 = v± (a) on a, u3 ^ f (a) on a, ^3 = v2 (a) on a, ^ = gr (ft) on 6, #!_ = Uj (j8) on /3, ^2 ~ 0 CO on 6, v% — u2 (/3) on /3, #g -~ r/ (6) on 6, ?>3 — ?/3 (/3) on /3, Writing ws. — fuL -f- (?/2 — ?^j) |- (w3 — w2) -h ... (uh — us^i), our object now is to show that as .9 -> oo the series for us and vs converge and represent potentials which are exactly the same in C. To establish the convergence of the series we shall make use of the following lemma. We note that ws = us — 11^^ is a logarithmic potential which is regular in A and which is zero on a. Let S,. (a) be its value at a point on a and let 83 be the maximum value of | 8S (a) \ . Now let </> be the logarithmic potential which is regular in A and which satisfies the boundary conditions (/» -= 1 on a, 0 — 0 on a. As the point (x, y) approaches one of the points of discontinuity m, <f> tends to a value 9 such that 0 < 6 < 1. Now a regular potential function attains its greatest value in a region on the boundary of the region, therefore <f> < 0 for all points of A and so there is a positive number e between 0 and 1 such that, on /?, <f> < e < 1 . Now ws -f Ss,<£ is zero on a and positive on a and is a logarithmic potential regular in A . Its least value is therefore attained on the boundary of A and so ws f £<.(/> > 0 within A. This inequality may be written in the f°rm o/i \ , . * n os (<p — €j ~f- w\ -!- to, > 0, and since </> < e on /3 it follows that ws 4 e8s > 0 on j8. In a similar way we can show that ivs — 8s<f> < 0 in A and so we may conclude that ws — e8s < 0 on /?. Combining the inequalities we may write The number e was derived from the function <f> associated with A. In a similar way there is a number T? associated with the region B and the curve a. Let K be the greater of these two numbers if the two numbers are not equal. Schwarz*s Alternating Process 249 Writing ts = vs ~ vs^ and using the symbol rs (/3) to denote the value of ts on p we use ra to denote the maximum value of | rs (/3) | on /?. We then find in a similar way that , M , K I < TS, and so we may write | Wa | < K8S1 | t, < KTS. We thus obtain the successive inequalities I «'2 ~ *'i I '-" I Therefore r2 \ on «. <j - - j i Therefore S^ < /cr2 < *2S2, ... S,+2 < *2sS2, T3 < *S3 < K2T2, ... Ts+2 < *2*T2. The series for ?/, and vs thus converge uniformly at all points of the boundary of C and so by Harnack's theorem represent regular logarithmic potentials which we may denote by ?/ and v respectively. Since ?/,, - v^^ on a and us = vs on /3 it follows that u -= v on the boundary of 6" and so u = v throughout C. Since, moreover, the series for u converges uniformly on the boundary of A and the series for v converges uniformly on the boundary of B these series may be used to continue the potential function u — v beyond the boundary of G into the regions A and B, and the potential function thus defined will have the desired values on a and 6. § 3-61. Flow round a circular cylinder. To illustrate the use of the com- plex potential in hydrodynamics we shall consider the flow represented by a complex potential % which is the sum of a number of terms y —— "* Xl - U (z + a2/z), X2 = M log z, ^ = ™ log ~° , z -- zt • , i 2 -- 22 X4= -1C log 2. ^ ^3 Writing z = re^, ^ = ^> 4- i^ we consider first the case in which x = Xi and £7 is real. We then have u~- iv= dx/dz = T7 ( 1 - a2/z2) , ^ = C7 sin 0 (r - a2/^)- The stream-function </r is zero on the circle r2 = a2 and also on the line y = 0. There is thus one stream-line which divides into two parts at a point S where it meets the circle ; these two portions reunite at a second point Sf on the circle and the stream -line leaves the circle along the line y = 0. Since z2 = a2 at the points S and 8' these points are points of stagnation (u = v = 0). It will be noticed that the stream-line y = 0 cuts the boundary r2 = a2 orthogonally. This is in accordance with the general theorem of § 1-72. 250 Two-dimensional Problems At a great distance from the circle we have u — iv = U, fy = C/y, and so the stream-lines are approximately straight lines parallel to the axis of x. Our function i/j is thus the stream -function for a type of steady flow past a circular cylinder. This flow is not actually possible in nature, the observed flow being more or less turbulent while for a certain range of speed depending upon the viscosity of the fluid and the size of the cylinder, eddies form behind the cylinder and escape downstream periodically* in such a way as to form a vortex street in which a vortex of one sign is almost equidistant from two successive vortices of the opposite sign and each vortex of this sign is almost equidistant from two successive vortices of the other sign. Vortices of one sign lie approximately on a line parallel to the axis of x and vortices of the other sign on a parallel line. Some light on the formation of this asymmetric arrangement of vortices is furnished by a study of the equilibrium and stability of a pair of vortices of opposite signs which happen to be present in the flow round the circular cylinder. The flow may be represented approximately by writing X = Xi + X* + *4> and choosing 20, zl9 z2, z3 so that the circle r2 = a2 is a stream -line, This condition may be satisfied by writing ItA,B,C,D are the points specified by the complex numbers z0 , zl , z2 , z3 , respectively, these equations mean that B is the inverse of A and 0 the inverse of D. In the theory of Helmholtz and Kelvin vortices move with the fluid. When the vortices are isolated line vortices this result is generally replaced by the hypothesis that the velocity of any rectilinear vortex to is equal to the resultant of the velocities produced at its location by all the other vortices which together with o> produce the resultant flow at an arbitrary point. In using this hypothesis the uniform flow U is supposed to be produced by a double vortex at infinity and the complex potential Ua2/z is interpreted as that of a double vortex at the origin of co-ordinates 0. The vortex at A will be stationary when Taking for simplicity the cavse when r2 = r0, ft2 = — 00, c' = c, and * Th. v. lUrmAn, Odtt. Nachr. p. 547 (1912); PJiys. Zeits. p. 13 (1912). The vortices have been observed experimentally by Mallock, Proc. Roy. Soc. London, vol. ix, p. 262 (1907); and by Benard, Comptes Rendus, vol. CXLVJI, pp. 839-970 (1908); vol. CLVI, pp. 1003-1225 (1913); vol. OLXXXII, pp. 1375-1823 (1926); vol. CLXXXIII, pp. 20-184 (1920). Cylinder and Isolated Vortices 251 separating the real and imaginary parts of the expression on the right, after multiplying it by 20 , we obtain the equations 0 = U (r0 - a2rQ~l) cos 00 - ±c cot 0Q + a*crQ2£l-1 sin 200 , 0 = U (r0 + a2r0-!) sin 00 - cr02/(r02 - a2) - \c + cr02 (r02 - a2 cos 200) Q-i, where £3 = r04 - 2a2r02 cos 200 + a4. The first equation gives 2UQ, sin 00 =- cr0 (a2 - r02), and when this value of U is substituted in the second equation it is found i/nat) 9 « i ci 9 • /i r02- a2 = ± 2r02sm00. This result was obtained by Foppl*, who also studied the stability of the vortices. The result tells us that the vortex can be in equilibrium if AB = AD. To confirm this result by geometrical reasoning we complete the parallelogram BADE and determine a point N on the axis of y such that ON = AN. Let M be the point of intersection of BC and AN , G the point of intersection of AC and BD. On the understanding that all lines used to represent velocities are to be turned through a right angle in the clockwise direction the velocities at A due to the different vortices may be represented as follows : Those due to the vortices at B and D by c/AB and c/AD respectively. Since AB = AD these two velocities together may be represented by c.AE/AB* along AE. The velocity due to the vortex at G may be represented by c/CA along GA and equally well by cGA/AB* along GA. The resultant velocity at A due to the vortices at B, C and D may thus be represented by c.GE/AB2 along OE. On the other hand, the velocity U is represented by U along ON, and the velocity due to the double vortex at O by U .NM/ON along NM. The velocity in the flow round the cylinder in the absence of the vortices is thus represented by U .OMJON. A A A Now MAE = ME A = OCM, therefore 0, M , A, C are concyclic and so 0&LC = OAC - OEG. This means that OM and EG are parallel. By choosing c so that c.EG/AB2= U.OM/ON the resultant velocity at A will H zero. Since the triangles ON A, OAD are similar, the equation for c becomes simply T7 OM AB2 T7 OM AB* TJ OM A n A ^ TT , c - U ON EG - U ON AG = U ON A ° = A C* ' U'a - U (r02 - a2) (1 - a4/r04)/a and implies that the strength of the vortex at A is greater the greater the distance of A from the origin. * L. Foppl, Munchen Sitzungsber. (1913). See also Howland, Journ. Roy. Aeron. Soc. (1925); M. Dupont, La Technique Atronautique, Dec. 15 (1926) and Jan. 15 (1927); W. G. Bickley, Proc. Roy. Soc. Lond. A, vol. cxix, p. 146 (1928). 252 Two-dimensional Problems The stream-lines in the flow studied by Foppl are quite interesting and have been carefully drawn by W. Miiller*. There are four points of stagnation on the circle, two of these, 8 and 8' ', lie on the line y = 0, while the other two, S0, $</, are images of each other in the line y = 0. Stream- lines orthogonal to the circle start at SQ and S0' and unite at a point T on the line y where they cut this line orthogonally. This point T is also a point of stagnation. Outside these stream -lines the flow is very similar to that round a contour formed from arcs of two circles which cut one another orthogonally; within the region bounded by these stream-lines there is a circulation of fluid and the flow between T and the circle is opposite in direction to that of the main stream. The stream-lines are, indeed, very similar to those which have been frequently observed or photographed in the case of the slow motion round a cylinderf. Let us now consider the case when there is only one vortex outside the cylinder and a circulation round the cylinder. We now put A Al ' A2 ' A3 * In this case u - w ={7(1- a*/z2) -f ik/z -f ic [(z - rQetd»)~l - (z - v^aVo)-1], and the component velocities of the vortex A are given by while for its image B ?/! -f ii\ -= — a2 (UQ — ivQ) r0~2e2lV If X, Y are the components of the resultant force on the cylinder per unit length, we have X + iY --= - Jpa P* (w2 + v2 -f- 2 -£} e"dO - (Xg + iYg) -I- (X* + iY+), say. J o V v* / Now when z — aet09 u" + v2 = 4C/2 sin2 8 + k*fa2 4- c2 (r02 - a2)2/a27?4 -I- 4C7 sin 0.[*/a - c (r02 - a2)/a.R2J - 2kc (r02 - a2)/a2J?2, where 7^2 = a2 + r02 - 2ar0 cos (/9 - 00). Therefore ^ + iYq = 2-rrip {kU — c (UQ -f- iv0)}. We have also for r — a * Znts.fur techmsche Physik, Bd. vm, S. 62 (1927); Mathematische Stromungslehre (Springer, Berlin, 1928), p. 124. t See especially the photographs published by Camichei in La Technique A fronautique, Nov. 15 (1925) and Dec. 15(1925). Circulation round a Cylinder and Vortex 253 therefore \*'W , . .-9f| n * a/ a<£ . 9</r a* + * dt 2rr gj, « - aet<? <i# == Trie (Uj + ivj) ~ 7rica2rQ~2 (UQ — iv0) e2*eo — 2Tric (t^ -f ivj). o ot Combining these results we have X + iY = 27rip {/J?7 — c (w0 + i#0) -f c (^ + i#i)}. This result may be extended to the case in which there are any number of vortices outside the cylinder*, the general result being -X" -f iY = 2 Trip \kU — 2 cs (?/2s 4- iv2s — In the special case when there is only one vortex and k = 0, 00 = 0, we have ut -f iz^ = — a2rQ~2 (UQ ~ ivQ), „, MJ TT /I n2r — 2\ ^>r /r 2 /t2\-l U-Q frt/Q \J ^1 U/ f Q f t/Ll Q \' Q ^ / > Z f iF = 27rpc [c/rQ — iU (I ~ a4r0~4)]. Introducing the coefficients of lift and drag, defined by X ~ p8U2.CJ)9 Y = pSU2.CL, 8 being the projected area, we find CD = (c/aU)2 7ra/r0, CL = - TT (c/aC7) (1 - a*r0-4). These results were obtained by W. G. Bickleyf who plots the lift-drag curves for r - 2a, 4a and 6a, and compares them with the published curves for Flettner rotors (rotating cylinders with end plates). With the last two values the agreement is fair except for low values of the lift. The stream-lines for the case of a single vortex outside the cylinder have been drawn by W. MiillerJ. EXAMPLES 1. If in a type of flow similar to that considered by Foppl the vortices at 20 and z2 are not images of each other in the line y = 0, one of the conditions that the vortices may be stationary in the flow round the cylinder is (r0 - a2rQ~l) cos 00 - (r2 - a*r2~l) cos 0a. 2. If in Foppl's flow the vortices move so that they are always images of each other in the line y — 0 the resultant force on the cylinder is a drag if 4r0* sin2 00 > (r02 - a2)2. [Bickley.] * H. Bateraan, Bull. Amer. Math. Soc. vol. xxv, p. 358 (1919); D. Riabouchinsky, Comples Rendus, t. CLXXV, p. 442 (1922); M. Lagally, Zeits. f. angew. Math. u. Mech. Bd. n, 8. 409 (1922). In this formula the even suffixes refer to the vortices outside the cylinder and the odd suffixes to the image vortices inside the cylinder. t Loc. cit. ante, p. 251. t Loc. cit. ante, p. 252. 254 Two-dimensional Problems 3. A plate of width 2a is placed normal to a steady stream of velocity U and vortices form behind the plate at the points Prove that the conditions are satisfied by = *. 8 (*» + «*)* -(*•» + «')* Prove also that when - *0 + 2 ( V + a2)*, the velocity does not take infinite values at the edges of the plate and the vortices are stationary. [D. Riabouchinsky.] § 3-71. Elliptic co-ordinates. Problems relating to an ellipse or an elliptic cylinder may be conveniently solved with the aicf of the sub- stitution . . U /<" , • \ 1 Y x + ly = c cosh (f + 2,7?) = c cosh f , which gives x = c cosh £ cos 77, T/ = c sinh £ sin 77. The curves £ = constant are confocal ellipses, 52 + j£__ = l c2 cosh2 f c2 sinh2 £ 5 the semi-axes of the typical ellipse being a = c cosh ^ and 6 = c sinh £. The angle 77 can be regarded as the excentric angle of a point on the ellipse. The curves 77 = constant are confocal hyperbolas, the semi-axes of the typical hyperbola being a' — c cos 77 and 6' = c sin 77. The first problem we shall consider is that of the determination of the viscous drag on a long elliptic cylinder which moves parallel to its length through the fluid in a wide tube whose internal surface is a confocal elliptic cylinder*. Considering a cylindrical element of fluid bounded by planes parallel to the plane of xy and a curved surface generated by lines perpendicular to this plane, the viscous drag per unit length on the curved surface of the cylinder is taken round the contour of the cross-section, w being the velocity parallel to a generator and p being the coefficient of viscosity. If the fluid is not being forced through the tube under pressure the pressure may be assumed to be constant along the tube and so in steady. motion the total viscous drag on the cylindrical element must be zero. * C. H. Lees, Proc. Roy. Soc. A, vol. xcn, p. 144 (1916). Viscous Resistance to Towing 255 Transforming the line integral into an integral over the enclosed area, we obtain the equation d*w Shu fo2 + ay2 ~ ' The boundary conditions are w = 0 when £ = ^ , and w = v when £ = £2 » we therefore write Since T? varies from 0 to 2-rr in a complete circuit round the contour of the cross-section, the total viscous force per unit length of the cylinder is 27TfJLV __ 2-JT^JLV S^=r^2 ~ log (aT+ b~] T~Togr(a2 "+ 62) ' If the inner ellipse reduces to a straight line of length 2c, the total drag • on the plane is D per unit length, where D [log (a, + 6,) - log (2c)] = 2^, and the resistance per unit area at the point x is (D/27r) (c2- z2)-*. It is clear from this expression that the resistance per unit area, i.e. the shearing stress, is much greater near the edges of the strip than near its centre line. The foregoing analysis may be used with a slight modification to determine the natural charges on two confocal elliptic cylinders regarded as conductors at different potentials. If V is the potential at (f, 77) and V = 0 for £ = £x , V = v for f = £2 , we have y & - &) = » (& - a and the density of charge on the cylinder £ = ^ is i aF3£ i _ ~ 477 8£ 3/1 -~ 47T 87 35 ^ ST(^ - &) c When the inner cylinder reduces to the strip whose cross-section is S^^ we have, when v = 2 (£x — ^2), CTj = (1/277-C) COS6C T^u and if, moreover, the outer cylinder is of infinite size ol becomes the natural charge on the strip when the total charge per unit length is equal to unity; this is the charge density on each side of the strip. To find the stream-function for steady irrotational flow round an elliptic cylinder when there is no circulation round the cylinder, we write ^ = ^i -f- 02 > where ^i ^ Uy ~ Vx = c (U sinh ^ sin 77 — V cosh £ cos 77) is the stream-function. for the steady flow at a great distance from the 256 Two-dimensional Problems cylinder and ^2 is the stream-function for a superposed disturbance in this flow produced by the cylinder. To satisfy the boundary condition </> = 0 at the surface of the cylinder, and the condition that the component velocities derived from </f2 are negligible at infinity, we write 02 = e~* (A cos ?; -f B sin ??). Choosing the constants so that tft — 0 on the cylinder, we have e-fi A = cV cosh & , e~fi B = — cU sinh & = ^7 =-6tZ7 where al5 6X are the semi-axes of the ellipse £ = &. We have also «!-{-&! = c^-^i, ax — fej_ = Cjg-fi Therefore 02 = (ax + 6^* (% — 6^"* e~f (7% cos T? — f/fij sin TJ), <f> + i*fi= (U - iV) z -f (*//>! + iF«j) (aj + 6^4 (% - &!)-* e-f. To find the electrical potential of a conducting elliptic cylinder which is under the influence of a line charge parallel to its generators, we need an expression for the logarithm log (z0 - z) =-- log [c (cosh £0 - cosh £}] - £0 -{- log Jc + log (1 - e^-^o) (l - e~^-^o) - & + logic -2 S w^e- Writing this equal to (/>0 -f ifa we have 00 ^o "-^ ^o -H l°g ^° — 2 S n-1e~nfo (cosh nf cos nrj cos >i-i + sinh ng sin ?^r; v^in nrjQ). To obtain a potential which is constant over the elliptic cylinder f ^ f i > wo write </> == ^0 4- ^ , where 00 ^>1 — £ (^rlne~"£ cos Tir; -f Bne~^ sin w^). 71-1 Each term of this series is indeed a potential function which vanishes at infinity. Choosing the constants An , Bn , so that the boundary condition <£ --= 0 on £ = ^ is satisfied by <f> = <f>0 + <f>l9 we have nAne~n%i = 2e~n^o cosh T^ cos TIT^Q, nBne~nti = 2e~n^o sinh n^ sin n^. Hence when ^ < ^ < £0, ^ ~ f o + ^°g lc + 2 7i-1en(^i"^o> sinh TZ- (^ — g) n~ I Summing the series we find that .-.«-J» eL Induced Charge Density 257 The corresponding stream-function is * - , + tan- -_ IL —.+ tan- r ' 1 — e*-fo cos (770 — ??) and when ^ = 0 the value of — for £ = 0 is __ _ __ cosh £0 - cos (770 - 77) ' The surface density of the charge on the plate £ = 0 is thus 1 dri [i _ ___ sinh/p 1 4:77 ds L cosh £0 — cos (770 — 77)] ' and the total charge is zero. When the total charge per unit length is 1, and the total charge per unit length of the line is — 1, the surface density of the charge on the cylinder is 1 drj sinh £0 , 27T ds cosh £0 — cos (779 — 77) " This is what C. Neumann* calls the induced charge density or the induced loading; it represents, of course, the charge on one side of the plate £ = 0. We shall write this expression in the form <r(0, 77; £)>??o) ^^TT^fe /S^°'7?; ^°'7?0^ and shall use a corresponding expression <*(fi>ih; £o»?7o) = 27T ~dss (£i>*?i; £o> ^o)* in which <* (t «'£ »\ sinh (|o - ^) . s (&, ih, 6. %) = cosh ^--gy-.-ooa & - ^)' ...... (A) and or (fj, 77!; ^0, 770) is the density of the induced charge for the elliptic cylinder f = ^ . Let us now consider the problem in which a function V is required to satisfy the condition V = / (77!) on the cylinder £ = &, while V is a regular potential function outside the cylinder £ = ^ but not necessarily vanishing at infinity. Some idea of the nature of the solution may be obtained by first considering the two cases Since ^ cos mr}9 V ~ e~™<£-£i> cos W77, ~ sin 77177, V — e-"*(£-£i> sin 77177. (&, %; f , 1?) - 1 + 2 S e-«««i> cos m (77 - 77,), w-l * Leipzig. JBer. Bd. LXH, S. 87 (1910). 17 258 Two-dimensional Problems the solution is given in these cases by the formula V = -S (&, %; & itf/fa) Ah ...... (B) and we may write o- (&, *h; £, i?) = o- (£1, i?i) 5 (&, ^; f, 77), ...... (C) where o>(^1, 77^ is the natural density per unit length when the total charge per unit length on the cylinder f = ^ is unity. This is a particular case of a general theorem due to C. Neumann*, which tells us that the density of the induced charge for a cylinder whose cross-section is a closed curve can be found when the natural density on- the cylinder and the corresponding potential is known. The expression for the induced charge is then of the form (C), where £ and 77 are conjugate functions such that f = constant are the equipotentials and 77 = constant, the lines of force associated with the natural charge. The undetermined constant factor occurring in the expressions for functions £ and 77 which satisfy the last condition should be chosen so that 77 increases by 2?r in one circuit round the cross-section of the cylinder. The formula (A) gives a potential which satisfies the conditions of the problem for a wide class of functions and for this class of functions we have the interesting relation 27r f M - Urn f ^ Sinh (^ ~~ &)/ foi) *h (f^f\ 2w/(,) - hm Jo cogh (f _ ^ _ cog (i-—j (f > £). The question naturally arises whether the function F given by (B) is the only function which fulfils the conditions of the problem. To discuss this question we shall consider the case when the ellipse f = f x reduces to the line £ = 0, i.e. the line S1S2. It will be noticed that when f (rj) = 1 the formula (B) gives 7=1. Now the potential <f> which is the real part of the expression (f> + i*ft=z (z2 - c2)~i - coth £, satisfies the condition that </> = 0 on the line £ = 0 and </> = 1 at infinity. Furthermore, the function <f>lt which is the real part of <£i -f i^i = c (z2 — c2)"^ = cosech £, satisfies the conditions </»! = 0 when £ = 0, fa = 0 when £ = oo. Hence a more general potential which satisfies the same conditions as v is TT j i -.-^ * F+ A<f> + B<f>l9 * Leipzig. Ber. Bd. LXII, S. 278 (1910). Munk^s Theory of Thin Aerofoils 259 where A and B are arbitrary constants. Now sinh £ ^ 1 4- e-*+t*o cosh £ — cos (T? — 770) ~~ 1 — e-£+<7>o __ o sinh $ -f i sin T/O cosh £ — cos 170 * Hence, if * fsinh £ -f i sin 770 cosh £ — A If* f - 2 TT J _„ [ - , - , — -- -T— 2 TT J _„ [ cosh £ - cos 770 smh £ ...... (D) where?/ and F are constants, the potentials u and v are conjugate functions which can be regarded as component velocities in a two-dimensional flow of an incompressible in viscid fluid. These component velocities satisfy the conditions u — U, v = F at infinity, v — / (77) on the line $x$2. This result is of some interest in connection with Munk's theory of thin aerofoils. In this theory an element ds of a thin aerofoil in a steady stream of velocity U parallel to Ox is supposed to deflect the air so as to give it a small component velocity v = u -~ in a direction parallel to the Ci XQ axis of y. Assuming that u = U -f- c, where e is a small quantity of the same order of smallness as yQ and dy0/dx0 , we neglect e , - , as it is of order CiXQ e2, and write v = U -~- . This is now taken to be the ^-component of (LXq velocity at points of the line S1S2 and the corresponding component velocities (u, v) for the region outside the aerofoil may be supposed, with a sufficient approximation, to be given by an expression of type (D). In this expression, however, the coefficient A is given the value 1 so that the velocity at the trailing edge will not be infinite. Now when | £ | is large we may write (cosh £ - cos Tjo)-1 = sech £ -f- cos TJQ sech2 £ + sinh £ = cosh £ — | sech £ — J sech3 £ -f- ... , cosech £ = sech £ -f J sech3 £ -f ____ Hence, when V = 0 the flow at a great distance from the origin is °f tyPe V+iu= iU + ft/* + &22 4 ... , c fir where fa = 2^ j ^ ( 1 -f cos 770 -f i sin rjQ) f fa) drjQ , c2 (n & = 2- J _ (* sin 770 cos 770 - sin2 770) / (770) <fy0 , 17-3 260 Two-dimensional Problems and by Kutta's theorem the lift, drag and moment per unit length of the aerofoil are given by the expressions of § 4- 71 L + iD = \p [(v+ iu)2 dz^ M = \pR f (v -f w)2 zdz = - Therefore L - - PcU2 f* (1 + cos T?O) J?y° cfy0, J - 7T ^#0 sin =0, These are the expressions obtained by Munk* by a slightly different form of analysis. A more satisfactory theory of thin aerofoils in which the thickness is taken into consideration, has been given by Jeffreys and is sketched in § 4-73. Since dxQ = — c sin f]QdrjQ, XQ = c cos T?O, we may write L - - 2pU2 [ (c + XQ) (c2 - x^ -^ dxQ J -c uX0 »c = 2pcU2\ (c + x0)-i(c-a;0Hy0^0, J ~C M = 2pU2 I (c2 - ^02)-* xyodx0f. J -c § 3-81. Bipolar co-ordinates. Problems relating to two circles which intersect at two points S1 and $2 with rectangular co-ordinates (c, 0), (— c, 0) revspectively may be treated with the aid of the conformal transformation . . ., . . v z = ic cot |^, ...... (A) where z ^ x + iy, £ = £ 4- ^ and (^, t/), (f , 7^) are the rectangular co- ordinates of two corresponding points P and TT. We shall say that the point P is in the z-plane and the point TT in the £-plane. The transformation may be said to map one plane on the other. It is easily seen that z - c where = (x _ c)2 -f s/2 - I z - c j 2 - 2cMe~\ = (a: + c)2 -f 7/2 = | z + c |2 = -... cosh 77 — cos f * National Advisory Committee for Aeronautics, Report, p. 191 (1924); see also J. S. Ames, Report, p. 213 (1925) and C. A. Shook, Amer. Journ. Math. vol. XLvm, p. 183 (1926). Bipolar Co-ordinates 261 The curves £ = constant are clearly circles through the points 8l and S2, while the curves 77 = constant p are circles having these points as inverse points. The two sets of curves form in fact two orthogonal systems of circles, as is to be ex- pected since the transformation (A) is conformal, and the corresponding sets of lines are perpendicular. S2 The expressions for x and y in Fis- 1()- terms of f and rj are x = M sinh 77, y = M sin f . At a point P0 of the line ^$2 we have £ = TT, therefore #0 - c tanh and the natural loading for this line is o-0= (I/we) cosh (iyo/2 )• The loading induced by a charge — 1 at the point P (f , r?) is, on the other hand, _ ° 7T0 [cosh (77 — 7?0) -f cos £] cosh fo/2) + cosh [fo/2) - 770] f — CTn | , > ,. UUO rt • cosh (Y] — T?O) -f cos f 2 EXAMPLE A potential function v is regular in the semicircle y > 0, x* -f y2 < a2 and satisfies the boundary conditions v = A when t/ =- 0, ^ -= .B when x2 + y2 — a2, prove that /(JO v = A-A' Jo (£ TT) dm m , OWTT cosh- where x -f t/y = ia cot j (£ -f *„), .4' = a#. § 3-82. Effect oj a mound or ditch on the electric potential. Let us now consider the complex potential 2c £ where K is a real constant at our disposal. The potential <f> is zero when £ = 0, for in this case ^ becomes cot , and is a purely imaginary K. /C quantity. It is also zero when £ = |/CTT, for then # = tan ^ , and is again a purely imaginary quantity. The potential <£ is thus zero on a continuous line made up ,of the portion of the line y = 0 outside the segment S1S2 and of the circular arc through S1S2 at points of which 262 Two-dimensional Problems S1S2 subtends the angle \KTT. Thus the complex potential x provides us with the solution of an electrical problem relating to a conductor in the form of an infinite plane sheet with a circular mound or ditch running across it. <,. d(f> (cosh r) — cos f )2 dc/> dy cosh 77 cos £ — 1 9f ' we find that on the axis of y, where 77 = 0, Al fy 2C of Also -£ = - -„ cosec2 -, d£ /c2 AC consequently the potential gradient on the axis x = 0 is 2 £ 2 cosec2 - (1 — cos |). At the vertex where £ = |KTT it is 2 (1 — cos I/CTT). On the plane y = 0, we have f - 0, and the gradient is „ cosech2 -.(cosh -n — I). K* K As ?? -vQ, x->co and the gradient tends to the value 1 which will be regarded as the normal value. As 77 -> ± oo, x -> ± c and the gradient tends to become zero or infinite according as K ^ 2. Fig. 17. Fig. 18. When K — 1 we have a semicircular mound. The gradient on the line x = 0 is everywhere greater than the normal value, at the vertex it is 2, and at a point at distance 2c above the vertex it is 10/9. When AC — 3 we have a semicircular ditch. The gradient on the line x = 0 is everywhere less than the normal value, at the bottom of the ditch it is 2/9 and at a point (0, c), at distance 2c above the bottom, it is 8/9. By making K -> 0 we obtain values of the . CL7T v cosec2 Electrical Effects of Peaks and Pits 263 gradient for the case of a cylinder standing on an infinite plane. We must naturally make c -*• 0 at the same time, in order to obtain a cylinder of finite radius a. The appropriate complex potential is X = <f> + i$ = an cot --- . . ...... (C) On the line x = 0 the potential gradient is CL7T\ , yr and tends to the normal value as y -> ± oo. When T/ == 2a the gradient is , which is nearly 2-5. At a distance 2a 2 above the summit, y = 4a and the gradient is ^ = 1-2337. On the axis 8 of x the gradient is 0 0 a7T i**fa7r\ — 0 cosech2 x2 \ x J As x -> 0 the gradient diminishes rapidly to zero, consequently the surface density of electricity is very small in the neighbourhood of the point of contact. EXAMPLE Fluid of constant density moves above the infinite plane y = 0 with uniform velocity U. A cylinder of radius a is placed in contact with the plane with its generators perpendicular to the flow. Prove that the stream function is derived from a complex potential of type (C) multiplied by U and calculate the upward thrust on the cylinder. [H. Jeffreys, Proc. Camb. Phil. Soc. vol. xxv, p. 272 (1929).] § 3*83. The effect of a vertical wall on the electric potential. Let h be the height of the wall, x = </> + ty the complex potential. If a is a constant, the substitution ., i m az = ih (a2 + x) ...... (**) makes the point z = ih correspond to x = 0> the points on the axis of x correspond to the points on </> = 0 for which 02 > a2, while the points on the axis of y for which y < h correspond to the points on <f) = 0 for which 02 < a2. Hence, if </> be regarded as the electric potential, a con- ducting surface consisting of the plane y = 0 and the conducting wall (x = 0, y < h) will be at zero potential*. If (r, 0), (/, 6') are the bipolar co-ordinates of a point P relative to $, the top of the wall, and to AS', the image of this point in the plane y = 0, we have h*x2 = - a2 (z2 + h*)= - a2rr'ei(e+e'>. Therefore h<f> = a (rrr$ sin J (6 4- 0')> Jufi - - a (rr')i cos £ (0 + (9'). Therefore 2Afy2 - a2 {[(x2 - i/2 + A2)2 + 4x2y2]* - (x2 - y2 H- A2)}. * G. H. Lees, Proc. Roy. Soc. London, A, vol. xci, p. 440 (1915), 264 Two-dimensional Problems The equipotentials have been drawn by Lees from the equation y2 (1 + a2x2/h2cf>2) = h2 + x2 -f h2<f>2/a*. To determine the surface density of electricity we differentiate equation (D), then When a: — 0, z = it/, /^ = — ia (A2 — y2)^, and so The* surface density is thus zero at the base and infinite at the top of the wall. When y = 0, z = x, /># = — ia (h2 -f x2)^, and so As a; -> oo this tends to the value a/h which may be regarded as the normal value of the gradient. At a distance from the foot equal to h the vertical gradient is 0-707 times the normal gradient. The curve along which the electric field strength has the constant value F is -given by a2 (x2 + y2) - h2F2 [(x2 - y2 + h2)2 + that i8' where (7?, 0) are the polar co-ordinates of P with respect to the origin. The curves F = constant may be obtained by inversion from the family of Cassinian ovals with 8 and $' as poles, they are the equipotential curves for two unit line charges at S and $', and a line charge of strength — 2 at the origin 0. The rectangular hyperbola 7/2 - X2 - |/*2 is a particular curve of the family. This hyperbola meets the axis of y at a point where the horizontal gradient is equal to the normal gradient. The force is equal in magnitude to the normal gradient at all points of this hyperbola. At points above the hyperbola the force is greater than a/h, at points below the hyperbola it is less than a/h. The curves along which the force has a fixed direction are the lemnis- cates defined by the equation 0 + 0' — 20 = constant. Each lemniscate passes through, S and S' and has a double point at 0. It should be noticed that the transformation az - ih (a2 - & enables us to map the upper half of the £-plane on the region of the upper Effect of a Vertical Wall 265 z-plane bounded by the line y = 0 and the vertical wall x = 0, y < h. This transformation makes the points at infinity in the two planes corre- sponding points. It may be observed also that if a2 — £2 — r2e2ie, the angle 20 ranges from — TT to TT. Hence, since az = hr sin 6 -f ihr cos 0, hr cos 0 is never negative and so it is the upper portion of the cut z-plane Wjhich corresponds to the upper portion of the £-plane. If we invert the z-plane from a point on the negative portion of the axis of y we obtain a region inside a circle which is cut along a radius from a point on the circumference to a point not on the circumference. The upper half of the £-plane maps into the interior of this region. If, on the other hand, we invert from the origin of the z-plane, the cut upper half plane inverts into a half plane with a cut along the y-axis from infinity to a point some distance above the origin. The point at infinity in the £-plane now maps into the origin in the z-plane. CHAPTER IV CONFORMAL REPRESENTATION § 4§ 11. Many potential problems in two dimensions may be solved with the aid of a transformation of co-ordinates which leaves V27 = 0 unaltered in form. It is easily seen that the transformation £ = / (*> y)> i? = 9 (x> y) furnished by the equation l=g + ii, = F(x + iy) = F (z) possesses this property when the function F is analytic, because a function of £ which is analytic in some region F of the £-plane is also analytic in the corresponding region G of the z-plane when regarded as a function of z. In using a transformation of this kind it is necessary, however, to be cautious because singularities of a potential function may be introduced by the transformation, and the transformation may not always be one-to- one, i.e. a point P in the £-plane may not always correspond to a single point Q in the z-plane and vice versa. Let F be a function of f and 77 which is continuous (D, 1), then 37^373£ aFcfy 37^3731 dVdrj dx 3£ dx drj 3xJ dy dg dy 877 dy' These equations show that if the derivatives of. £ and 77 are not all finite at a point (x, y) in the z-plane, the derivatives ~— , ~ may be infinite 37 dV even though -^ and _ are finite. A possible exception occurs when 3F 37 yr and -x both vanish, i.e. at a point of equilibrium or stagnation. At any point (#0, yQ) in the neighbourhood of which the function F (z) can be expanded in a Taylor series which converges for | z — z0 | < c, we have F (z) = 1 an (z - z0)», n-O where z = x + iy, z0 = XQ + iy0 , and if £ = f + i, = J (z), £0 = £0 + irjQ = F (z0), we may write d£ - £ - £0 = F (z) - ^ (z0) = rfz [f (z) + e], where rfz = z — z0 and e -> 0 as dz -> 0. Hence d£ = dz.^1' (x) approximately. Properties of the Mapping Function 267 This relation shows that the (x, y) plane is mapped conformally on the (f , rj) plane for all points at which | F' (z) \ is neither zero nor infinite. We have in fact the approximate relations da = ds | F' (z) | , <f>= 6+ a, where dz = ds.eie, dt, = dv.e^, F' (z) = | F' (z) | 4*. These relations show that the ratio of the lengths of two corresponding linear elements is independent of the direction of either and that the angle between two linear elements dz, 8z at the point (x, y) is equal to the angle between the two corresponding linear elements at (£, 77). The first angle is, in fact, 6 — 6' ', while the second angle is <!>-<l>'=(0 + a)-(8' + a) = 0- 6'. These theorems break down if some of the first coefficients in the expansion £- ^^^an(z~z.Y (A) n-l are zero. If, for instance, a^ = a2 = ... am^ = 0, we have for small values Of | Z - ZQ | £ - £o = am (* ~ Zo)m> and the relation between the angles is , <f> = m0 + am, where am= \am\ e"m. This gives <f> - </>' - m (0 - 6'). More generally if there is an expansion of type (A) in which the lowest index m is not an integer a similar relation holds. § 4- 12. The way in which conformal representation may be used to solve electrical and hydrodynamical problems is best illustrated by means of examples. One point to be noticed is that frequently the transformation does not alter the essential physical character of the problem because an electric charge concentrated at a point (line charge) corresponds to an equal electric charge concentrated at the corresponding point, a point source in a two-dimensional hydrodynamical problem corresponds to a point source and so on. These results follow at once from the fact that if <f> + i$ is the complex potential we may write <t> + i*f>=f(x+ iy) - g (f + iy), and if <f> is the electric potential, the integral \difj taken round a closed curve is ± 47T times the total charge within the curve. Now the interior of a 268 Confonnal Representation closed curve is generally mapped into the interior of a corresponding closed curve and a simple circuit generally corresponds to a simple circuit, moreover 0 is the same in both cases and so the theorem is easily proved. It should be noted that a simple circuit may fail to correspond to a simple circuit when the closed curve contains a point at which the conformal character of the transformation breaks down. Another apparent exception arises when a point (x, y) corresponds to points at infinity in the (£, rj) plane, but there is no great difficulty if these points at infinity are imagined to possess a certain unity. In fact mathematicians are accustomed to speak of the point at infinity when discussing problems of conformal representation. This convention is at once suggested by the results obtained by inversion and is found to be very useful. There is no ambiguity then in talking of a point charge or source at infinity. We have s§en that certain angles are unaltered by a conformal trans- formation and can consequently be regarded as invariants of the trans- formation. Certain other quantities are easily seen to be invariants. Writing where d (x, ?/), d (£, 77) are elements of area in the two planes, we have J ( _L - r \ - ^ ^ _i_ W '"dr"'1 °" J Sx The quantities </> and 0 are usually taken to be invariants in a con- formal transformation and the foregoing relations indicate that * ffl"V9<A\2 /3^\21 7 ^ . and \\\(J} +U \dxdy cx2 y2/ }}[\dxj \dyj J * are invariants. In the theory of electricity the first integral is proportional to the total charge associated with the area over which the integration takes place. In hydrodynamics the second integral represents the total vorticity associated with the area and the third integral is proportional to the kinetic energy when the density of the fluid is constant. The invariant character of the integrals (udx -f vdy) and (udy — vdx) is easily recognised because these represent \d(f> and \difs respectively. Riemann Surfaces and Winding Points 269 § 4*21. The transformation w = zn. When n is a positive integer the transformation w = zn does not give a (1, 1) correspondence between the w-plane and the z-plane but it is convenient to consider an ti-sheeted surface instead of a single plane as the domain of w. For a given value of w the equation zn — w has n roots. If one of these is Zl the others are respectively Z2 = Z^, Z3 = Z^2, ... Zn = 21o>w-1, where oj = e2rrt/n. If 2 = reie we may adopt the convention that for Zly 0< 710 < 2,T, Z2, 2n< n9< 477, Zn, (2W — 2) 77 < 710 < 2/177. Defining the sheet (m) to be that for which Wm = Zmn we can say that W± is in the first sheet, PF2 in the second sheet, and so on. The n sheets together form a "Riemann surface" arid we can say that there is a (1, 1) correspondence between the z-plane and the Riemann surface composed of the sheets (1), (2), ... (n). If w --= Re1® we have 0 = n9, and so when w = W^n» we have (2m — 2) 77 < 0 < 2w7r. The z-plane is .divided into n parts by the lines joining the origin to the corners of a regular polygon, one of whose corners is on the axis of x. These n portions of the z-plarie are in a (1, 1) correspondence with the n sheets of the Riemann surface. The n lines just mentioned each belong to two portions and so correspond to lines common to two sheets. It is by crossing these lines that a point passes from one sheet to another as the angle 0 steadily increases. The point O in the w-plane is a winding point of the Riemann surface, its order is defined as the number n — 1. A circle | w — W \ = an corresponds to a curve | zn — Zn \ = an, which belongs to the class of lemniscates* ^2 ... rn = an, where rx , r2 , . . . rn are the distances of the point z from the points Zx , Z2 , . . . Zn which correspond to W. In the present case the poles of the lemniscate are at the corners of a regular polygon and the equation of the lemniscate can be expressed in the form r2n _ 2rnSn cos n (0 - 0) + R2n = a2n (reie = z, Rei& = Z). When n = 2 a circle in the z-plane corresponds to a lima^on. To see this we write w = u + iv, z = x + iy, then u = x2 — y2y v = 2xy. * This is the name used by D. Hilbert, Gutt. Nachr. S. 63 (1897). The name cassinoid is used by C. J. de la Vallee Poussin, Mathesis (3), t. n, p. 289 (1902), Appendix. The geometrical pro- perties and types of curves of this kind are discussed by H. Hilton, Mess, of Math. vol. XLVIII, p. 184 (1919), reference being made to the earlier work of Serret, La Goupilliere and Darboux. 270 Conformal Representation Hence, if . ,0 0 0 (x + a}* + y2 = c2, we have [u2 4- v2 - 2a2u + (a2 - c2)2]2 - 4c4 (^2 + v2), or, if /7 - u + c2 - a2, F = v, U ^ RcosQ, V ^ R sin 0, ([72 + 72 „ 2c2Z7)2 - 4a2c2 (?72 + F2), R = 2ac + 2c2cos0. EXAMPLES 1. The curve r2n — 2rncn cos n0 -f- cndn — 0 has w ovals each of which is its own inverse with respect to a circle centre O and radius \/(cd). The ordinary foci Bl9B2,...Bn invert into the singular foci A19 A2, ... An, the polar co-ordinates of Bs being given by r ^ d, nd — ZSTT. 2. Line charges of strength 4- 1 are placed at the corners of a regular polygon of n corners and centre 0, while line charges of strength — 1 are placed at the corners of another regular polygon of n corners and centre O. Prove that the equipotentials are w-poled lemnis- cates. [Darboux and Hilton.] 3. Prove also that the lines of force are n-poled lemniscates passing through the vertices of the regular polygons. § 4-22. The bilinear transformation. The transformation r ^ az±b (A} ^~ az+$> (A) in which a, 6, a, ]3 are complex constants, is of special interest because it is the only type of transformation which transforms the whole of the z-plane in a one-to-one manner into the whole of the £-plane and gives a conformal mapping of the neighbourhood of each point. If a i- 0 there are generally two points in the z-plane for which £ = z. These are given by the quadratic equation az2 + (j8 - a) z - b - 0. Let us choose our origin in the z-plane so that it is midway between these points, then /? = a and if we write b = ac2 the self-corresponding points are given by z = ± c. The transformation may now be written in theform £+_c._a + ca« + c £ — c a — caz ~ c ' From this relation a geometrical construction for the transformation is easily derived. Writing a + ca ^ ;„ £ + c = RI&*I, z + c - r^'i, £ — c = R2el®2, z — c = r2eiezr R r we have the relations n1 = p -1 , ^2 *> 0! - 0, = CO + (^ - *,). The Bilinear Transformation 271 If Sl and S2 are the self -corresponding points these relations tell us that a circle through Sl and S2 generally corresponds to a circle through Sl and $2, but in an, exceptional case it may correspond to a straight line, namely the line $!&,. Again, a circle which has Sl and S2 as inverse points corresponds to a circle which has $x and S2 as inverse points. By a suitable displacement of the z and £-planes we can make any given pair of points the self-corresponding points provided the self-corre- sponding points &re distinct, for if the displacements are specified by the complex quantities u and v respectively, the transformation may be written in the form and we can choose u and v so that the equation £ = z has assigned roots zl and z2. We may conclude from this that the transformation maps any circle into either a straight line or a circle; a result which may be proved in many ways. One proof depends upon the theorem that in a bilinear transformation of type (A) the cross-ratio of four values of z is equal to the cross-ratio of the four values of £; i.e. (z- z,) (*8 -_z3) (r^j^Hk - k) (* - z2) (*3 - *i) (£ - £2) (& - £iV Now the cross-ratio is real when the four points lie on either a straight line or a circle, hence four points on a circle must map into four points which are either collinear or concyclic. If in the transformation (A) we choose u so that au + ft = 0, and v so that av = a, the transformation takes the form y 1 9 ,._ £z=&2, ...... (B) where a2k2 — ab — aft. By a suitable rotation of the axes of reference we can reduce the transformation to the case in which k is real, and this is the case which will now be discussed. The transformation evidently consists of an inversion in a circle of radius k with centre at the origin followed by a reflection in the axis of x. The points z = ± k are self-corresponding points and if these are denoted by S1 and S2 it is easily seen that two corresponding points P and Q lie on a circle through S^ and S2. The figure has a number of interesting properties which will be enumerated. 1. Since z (£ -f k) = k (z -f k) the angles /SXPO, QS^ are equal, and so the angles S1PO, S2PQ are also equal. 2. The triangles S1PO, QPS% are similar, and so PSi.PSt~PO.PQ. 272 Conformal Representation 3. If C is the middle point of PQ we have CS^CS^ = CP2, also PQ bisects the angle S^CS2. The four points Sl9 P, S2, Q on the circle form a harmonic set. This follows from the relation 1 4- _ =-r z_ k^z + k z- V which is easily derived from (B). The angle PS^C is equal to the angle 8^0. The lines ^P, P0, (7^ thus form an isosceles triangle. In the case when the self-corresponding points coincide we have a - fl = 2ac, b = - ac2, where c is the self-corresponding point. The transformation may now be written in the form ~ = Q -"- •--+ — , £ + ac * 0. £— c ft + ac z ~ c r It may be built up from displacements and transformations of the type just considered and so needs no further discussion. The only other interesting special case is that in which the transformation then consists of a displacement followed by a magnification and rotation. § 4-23. Poissorts formula and the mean value theorem. Bocher has shown by inversion that Poisson's formula may be derived from Gauss's theorem relating to the mean value of a potential function round a circle. Let C and C' be inverse points with respect to the circle F of radius a, and let CO' = c. Inverting with respect to a circle whose centre is C' and radius c, the point 8 on the circle transforms into a point 8'. We shall suppose that C' is outside the circle F, then S is inside the circle F. Let dsy els' be corresponding arcs at 8 and 8' respectively and let the polar co-ordinates of C and C1 be (r, 6), (/, 0) respectively, where rr' = a2. The circle F inverts into a circle with centre C and radius given by the formula aa' - or, for rR „. PS' ~ r - r ° C£ -CC'S CM a — r cr , = c -,— - =— = a . r — a a Also r*C'S* - a*.CS2 - a2 [r2 -fa2- 2ar cos (a - 0)], where (a, a) are the polar co-ordinates of 8. Writing ds' ~ a'dv , we have , , _ c2ds ca2da __ rcda = a'T C"&» ^ r ". C'S* = a2 - 2ar cos~(a - "^"Tr5 ' and re = a2 — r2, consequently the formula of Poisson becomes Circle and Half Plane 273 This formula states that the mean value of a potential function round the circumference of a circle is equal to the value of the function at the centre of the circle. Hence Poisson's formula may be derived from this mean value theorem and is true under the same conditions as the mean value theorem. § 4-24. The conformal representation of a circle on a Jialf plane*. If two plane areas A and A± can be mapped on a third area AQ they can be mapped on one another, eonseqiiently the problem of mapping A on At reduces to that of mapping A and Al on some standard area A0. This standard area AQ is generally taken to be either a circle of unit radius or a half plane. The transition from the circle x2 -f y2 < 1 in the z-plane to the half plane v > 0 in the ?#-plane is made by means of the substitutions z = x -f iy, w = u -j- iv, z (i -j- w) = i — w, Dx - 1 - u2 - v2, l)y - 2i/, (I) where D = u2 {- (1 -f- v)2, 4/D - (1 4- x)2 -f y2. When v = 0, the substitution u = tan 6 gives x = cos 20, y = sin 29. As 26 varies from — TT to TT, the variable u varies from — oo to oo and so the real axis in the w-plane is mapped in a uniform manner on the unit circle x2 -f- y2 = 1 in the z-plane. Since, moreover, D (1 - x2 - y2) = 4?;, Dy = 2u9 we have v > 0 when x2 -f y2 < 1, consequently the interior of the circle is mapped on the upper half of the w-plane. When u and v are both infinite or when either of them is infinite, we have x = — 1, y = 0; hence the point at infinity in the w-plane corre- sponds to a single point in the z-plane and this point is on the unit circle. The transformation (I) may be applied to the whole of the z-plane; it maps the region outside the circle x2 + y2 = 1 on the lower half (v < 0) of the w-plane. A line y = mx drawn through the centre of the circle corresponds to a circle m (1 — u2 — v2) = 2u whjch passes through the point (0, 1) which corresponds to the centre of the circle, and through the point (0, — 1) which corresponds to the point at infinity in the z-plane. This circle cuts the line v = 0 orthogonally. Two points which are inverse points with respect to the circle x2 -j- y2 = 1 map into points which are images of each other in the line v = 0. * This presentation in §§ 4-24, 4-61 and 4-62 follows closely that given in Forsyth's Theory of Functions and the one given in Darboux's Thdorie generate des surfaces, 1. 1, pp. 170-180. B 18 274 Conformal Representation The upper half of the w-plane may be mapped on itself in an infinite number of ways. To see this, let us consider the transformation Y aw + b dt, — b /T1A £ = -7-J9 W = — =, (II) cw -f d a — c£ in which a, 6, c and d are real constants and £ = £ -f irj. When w is real £ is also real and vice versa, hence the real axes corre- spond. Furthermore, 7? [(cu + d)2 + c2v2] = (ad- be) v, hence Had — be is positive, T? is positive when v is positive. There are three effective constants in this transformation, namely, the ratios of a, b and c to d, hence by a suitable choice of these constants any three points on the axis of u may be mapped into any three points on the axis of £. In fact, if uly u2, us are the values of u corresponding to the values gl9 £2, £3 °f l> we can say from the invariance of the cross-ratio that (£ - li) (& ~ la) _ (w ~ ^i) fa? - ^a) (£ - sY) (& ~ ~ li) (^ -^2) fas ~ ^i) ' and so the equation of the transformation may be written down in the previous form, the coefficients being a = && fa2 ~ ^3) + fall faa - %) 4- Iil2 fai ~ ^2), & - ^3) + Wa^ila (la - li) + ^1^2 Is (li ™ I2)» |3) + ^2 (la ~ li) + ^3 (li ~ la)» - ^3) + 7^3^ (|3 - ^) + U1U2 (^ - f2). The quantity ad — 6c is given by the formula ad - be - (£2 - &) (& - ^) (^ - f2) (^2 - u3) (u^ - %) (^ - u2). If i^, u2, t63 are all different the coefficients c and d cannot vanish simultaneously, for the equations c = 0, d = 0 give b2 S3 S3 "" b 1 _ b 1 ~~ b 2 «1 ( V - %2) "" ^2 (^32 ~ %2) ~ ^3 (^l2 - ^22) ' and these equations imply that *i (W22 - %2) + ^2 (^32 ~ %2) + ^3 (%2 - ^22) = 0, or (w2 - wj) (i^ - Ul) (HI - ^2) = 0, if the quantities fx, ^2, ^3 are also all different. In a similar way it can be shown that a and c cannot vanish simultaneously and that a and b cannot vanish simultaneously. Poincar6 has remarked that the transformation (II) can usually be determined uniquely so as to satisfy the requirement that an assigned point £ and an assigned direction through this point should correspond to an assigned point w and an assigned direction through this point. The proof of the theorem may be left to the reader, Riemanrfs Problem 275 who should examine also the special case in which one or both of the points is on the real axis in the plane in which it lies. EXAMPLES 1. Prove that Poisson's formula 7 = 2* j _„. T-2rcos(6~-'iJ')~-rr* \r\<l maps into the formula of § 3*11, _ 1 f =° yF (x'} dx' ~ TT / -co (x -x')2 + y*' where / (0') = F (tan £0'). 2. Prove that the transformation maps the half plane y > 0 on the unit circle | w | < 1 in such a way that the point z0 maps into the centre of the circle. § 4-31. Riemann's problem. The standard problem of conformal repre- sentation will be taken to be that of mapping the area A in the z-plane .on the upper half of the w-plane in such a way that three selected points on the boundary of A map into three selected points on the axis of u. This is the problem considered by B. Riemann in his dissertation. The problem may be made more precise by specifying that the function / (w) which gives the desired relation z=f(w) should possess the following properties : (1) / (w) should be uniform and continuous for all values of w for which v > 0. If WQ is any one of these values/ (w) should be capable of expansion in a Taylor series of ascending powers of w — WQ which has a radius of convergence different from zero. (2) The derivative/' (w) should exist and not vanish for v > 0 ; indeed, if /' (w) = 0 for w = WQ there will be at least two points in the neighbour- hood of w0 for which z has the same value. This is contrary to the require- ment that the representation should be biuniform. (3) / (?#) should be continuous also for all real values of tv, but it is not required that in the neighbourhood of one of these values, WQ, the function / (iv) can be expanded in a Taylor series of ascending powers of w — WQ) for, as far as the mapping is concerned, / (w) is defined only for v > 0. (4) Considered as a function of z, the variable w should satisfy the same conditions as/ (w). If/ (w) satisfies all these requirements it will give the solution of the problem. The solution is, moreover, unique because if two functions , , . , . z = f(w), z = g(w) give different solutions of the problem, the transformation 18-2 276 Conformal Representation will map the upper half plane into itself in such a way that the points ^i > uz ) U3 m&p into themselves. Now it can be proved that a transformation which maps the upper half plane into itself is bilinear and so the relation between w and W is (w - uj (u2 - u3) ^ (W -7^) (w_2_- ?/3) (w - u2) (u3 - uj ' (W - u2) (w.a - u^ w — HI W — u, mr A A ui — — . — . . W — U2 W — U2 This reduces to (11, - W) (u, - u2) - 0, and so W = w. § 4-32. TAe (jeneral problem of conformed representation. The general type of region which is considered in the theory of conformal representation may be regarded as a carpet which is laid down on the z -plane. This carpet is supposed to have a boundary the exact nature of which requires careful specification because with the aim of obtaining the greatest possible generality, different writers use different definitions of the boundary curve. There may, indeed, be more than one boundary curve, for a carpet may, for instance, have a hole in its centre. For simplicity we shall suppose that each boundary curve is a simple closed curve composed of a finite number of pieces, each piece having a definite direction at each of its points. At a point where two pieces meet, however, the directions of the two tangents need not be the same; a carpet may, for instance, have a corner. The tangent may actually turn through an angle 2?r as we pass from one piece of a boundary curve to another and in this case the boundary has a sharp point which may point either inwards or outwards. It turns out that the fojmer case presents a greater difficulty than the latter. In special investigations other restrictions may be laid on each piece of a boundary curve and from the numerous restrictions which have beer used we shall select the following for special mention. (1) The direction of the tangent is required to vary continuously as a point moves along the curve (smooth curve*). (2) The curvature is required to vary continuously as a point moves along the curve. (3) The curve should be rectifiable, i.e. it should be possible to define the length of any portion of the curve with the aid of a definite integral which has a precise meaning specified beforehand, such as the meaning given to an integral by Riemann, Stieltjes or Lebesgue. A simple curve which possesses the first property may be called a curve (CT), one which possesses the properties 1 and 2 a curve (CTC), a curve which possesses the properties 1 and 3 may be called a curve (RCT). * A curve which is made up of pieces of smooth curves joined together may be called smooth bit by bit ("Stiickweise glatte Kurve"; see Hurwitz-Courant, Berlin (1925), Funktionentheorie). Properties of Regions 277 The carpet will be said to be simply connected when a cross cut starting from any point of the boundary and ending at any other divides the carpet into two pieces. A carpet shaped like a ring is not simply connected because a cut starting from a point on one boundary and ending at a point on the other does not divide the carpet into two pieces. ^Such a carpet may, however, be made simply connected by making a cut of this type. When we consider a carpet with n boundaries which are simple closed curves we shall suppose that the boundaries can be converted into one by a suitable number of cuts which will at the same time render the carpet simply connected. It will be supposed, in fact, that the carpet is not twisted like a Mobius' strip when the cuts have been made. Any closed curve on a simply connected carpet can be continuously deformed until it becomes an infinitely small circle. This cannot always be done on a ring-shaped carpet as may be seen by considering a circle con- centric with the boundary circles of a ring, and it cannot be done in the case of a curve which runs parallel to the edge of a singly twisted Mobius' strip formed by joining the ends of a thin rectangular strip of paper after the strip has been given a single twist through 180°. Such a closed curve is said to be irreducible and the connectivity of a carpet may be defined with the aid of the number of different types of irreducible closed curves that can be drawn on it. Two closed curves are said to be of different types when one cannot be deformed into the other without any break or crossing of the boundary. It is not allowed, for instance, to cut the curve into pieces and join these together later or in any way to make the curve into one which does not close. A simply connected carpet may cover the plane more than once; it may, for instance, be folded over, or it may be double, triple, etc. In the latter case it is called a Riemann surface, i.e. a surface consisting of several sheets which connect with one another at certain branch lines in such a way as to give a simply connected surface. When there are only two sheets it is often convenient to regard them as the upper and lower surfaces of a single carpet with a cut or branch line through which passage may be made from one surface to the other. In the case of a ring-shaped carpet we generally consider only the upper surface, but if the lower surface is also considered and a passage is allowed from one surface to the other across either one or both of the edges of the ring a surface with two sheets is obtained, but this doubly sheeted surface is not simply connected because a curve concentric with the two edges is still irreducible. In a more general theory of conform al representation the mapping of multiply connected siirf aces is considered, but these will be excluded from the present considerations . A carpet may also have an infinite number of boundaries or an infinite number of sheets, but these cases will also be excluded. When we speak of 278 Conformal Representation an area A we shall mean the right side of a carpet which is bounded by a simple closed curve formed of pieces of type (RCTC) and is not folded over in any way. The carpet will be supposed, in fact, to be simply con- nected and smooth, the word smooth being used here as equivalent to the German word "schlicht," which means that the carpet is not folded or wrinkled in any way. The function F (z) maps the circular area | z \ < 1 into a smooth region if F fa) - F (z2) £- 1— * °' whenever | zl \ < 1 and | z2 | < 1 . We shall be occupied in general with the conformal representation of one simple area on another, and for brevity we shall speak of this as a mapping. In advanced works on the theory of functions the problem of conformal representation is considered also for the case of Riemann surfaces and the more general theory of the conformal representation of multiply connected surfaces is treated in books on the differential geometry of surfaces. For many purposes it will be sufficient to consider the problem of conformal representation for the case of boundaries made up of pieces of curves having the property that the co-ordinates of their points can be expressed parametrically in the form where the functions / (t) and g (t) can be expanded in power series of type 2an(*-*0)n, ...... (HI) n-O which are absolutely and uniformly convergent for all values of the parameter t that are needed for the specification of points on the arc under consideration. In such a case the boundary is said to be composed of analytic curves and this is the type of boundary that was considered in the pioneer work of H. A. Schwarz, but the restriction of the theory to boundaries composed of analytic curves is not necessary* and a method of removing this restriction was found by W. F. Osgoodf. His work has been followed up by that of many other investigators^. In the power series (III) the quantities tQ, an are constants which, of course, may be different for different pieces of the boundary. There are really two problems of conformal representation. In one problem the aim is simply to map the open region A bounded by a curve * In the modern work the boundary considered is a Jordan curve, that is, a curve whose points may be placed in a continuous (1,1) correspondence with the points of a circle. t Trans. Amer. Math. Soc. vol. i, p. 310 (1900). t Particularly E. Study, C. Caratheodory, P. Koebe and L. Bieberbach. Exceptional Cases 279 a on the open region B bounded by a curve 6, there being no specified requirement about the correspondence of points on the two boundaries. In the second problem the aim is to map the closed realm* A on the closed realm B in such a way that each point P on a corresponds to only one point Q which is on b and so that each point Q on b corresponds to only one point P which is on a. It is this second problem which is of most interest in applied mathematics. If, moreover, in the first problem the correspondence between the boundaries is not one-to-one the applied mathematician is anxious to know where the uniformity of the corre- spondence breaks down. Existence theorems are more easily established for the first problem than for the second and fortunately it always happens in practice that a solution of the first problem is also a solution of the second ; but this, of course, requires proof and such a proof must be added to an existence theorem that is adapted only for the first problem. The methods of conf ormal representation are particularly useful because they frequently enable us to deduce the solution of a boundary problem for one closed region A from the solution of a corresponding boundary problem for another region B which is of a simpler type. When the function which effects the mapping is given by an explicit relation the process of solution is generally one of simple substitution of expressions in a formula, but when the relation is of an implicit nature or is expressed by an infinite series or a definite integral the direct method of substitution becomes difficult and a method of approximation may be preferable. A method of approximation which is admirable for the purpose of establishing the existence of a solution may not be the best for purposes of computation. § 4-33. Special and exceptional cases. It is easy to see that it is not possible to map the whole of the complex z-plane on the interior of a circle. Indeed, if there were a mapping function / (z) which gave the desired representation, / (z) would be analytic over the whole plane and | / (z) \ would always lie below a certain positive value determined by the radius of the circle into which the z-plane maps, consequently by Liouville's theorem / (z) would be a constant. A similar argument may be used for the case of the pierced z-plane with the point z0 as boundary. By means of a transformation z — z0 = 1/z' the region outside z0 can be mapped into the whole of the z'-plane when the point z' = oo is excluded. The mapping function is again an integral function for which \f(z')\<M, and is thus a constant. On account of this result a region considered in the mapping problem is supposed to have more than one external point, a point on the boundary being regarded as an external point. * We use realm as equivalent to the German word " Bereich, " and region as equivalent to " Gebiet." 280 Conformal Representation The next case in order of simplicity is the simply connected region with at least two boundary points A and B. If these were isolated the region would not be simply connected. We shall therefore assume that there is a curve of boundary points joining A and B. This curve may contain all the boundary points (Case 1) or it may be part of a curve of boundary points which may either be closed or terminated by two other end-points C and F. The latter case is similar to the first, while the case of a closed curve is the one which we wish eventually to consider. The simplest example of the first case is that in which the end-points are z -= 0 and z = oo, the boundary consisting of the positive x-axis. The region bounded by this line can be regarded as one sheet of a two-sheeted Riemann surface with the points 0 and oo as winding points of the second order, passage from one sheet to the other being made possible by a junction of the sheets along the positive #-axis. The whole of this Riemann surface is mapped on the z' -plane by means of the simple transformation z' — y'z, which sends the one sheet in which we are interested into the half plane 0 < 6' < 77, where z' = r'elQf. In the case when the boundary consists of a curve joining the points z = a, z =•- 6, these points are regarded as winding points of the second order for a two-sheeted Riemann surface whose sheets connect with each other along the boundary curve. This surface is mapped on the whole z'-plane by means of the transformation ' (Z ~ "}* cz' ~ 1 (z -~b) - cz - *> and in this transformation one sheet goes -into the interior, the other into the exterior of a certain closed curve (7. The mapping problem is thus reduced to the mapping of the interior of C on a half plane or a unit circle. Finally, by means of a transformation of type z= Az' + B, we can transform the region enclosed by C into a region which lies entirely within the unit circle | z \ < 1 and our problem is to map this region on the interior of the unit circle | £ | < 1 by means of a transformation of type J=/(z). § 4-41. The mapping of the unit circle on itself. If a and a are any two conjugate complex quantities and a is a real angle the quantities e*a — a and 1 — dela have the same modulus, consequently if )3 is another real angle the transformation (1 -&) £ = e*(z-a) (A) maps | z | = 1 into | £ | = 1, and it is readily seen that the interior of one circle maps into the interior of the other. It should be noticed that this Circle Mapped on Itself 281 transformation maps the point z = a into the centre of the circle | £ | = 1, If we put a = 0 the transformation reduces to the rotation £ = ze*, which leaves the centre of the circle unaltered. If we can prove that this is the most general conformal transformation which maps the interior of the unit circle into itself in such a way that the centre maps into the centre it will follow that the formula (A) gives the most general trans- formation which maps the unit circle into itself. The following proof is due to H. A. Schwarz. Let / (z) be an analytic function of z which is regular in the circle | z | = 1 and satisfies the conditions \f(z)\< 1 for \z\< 1, /(0) = 0. If "<£(*) =/(*)/*> </>(<>) =/'(0), the function </> (z) is also regular in the unit circle, and if | z \ = r, where r < I,1 we have I <£ (z) I < l/r- But since </> (z) is analytic in the circle | z \ = r the maximum value of | <f> (z) | occurs on the boundary of this region and not within it, hence for a point z0 within the circle | z \ = r, or on its circumference, we have the inequality | <f> (z0) | < l/r (Schwarz's inequality*). Passing to the limit r -> 1 we have the inequality | <£ (z0) | < 1 for | z0 | < 1. Now let £ = / (z), z -= g (£) be the mapping functions which map a circle on itself in such a way that the centre maps into the centre, then by Schwarz's inequality | £/z | < 1, | z/£ | < 1. Rence | £/2 | =•= 1, and so | </> (z) \ is equal to unity within the unit circle. Now an analytic function whose modulus is constant within the unit circle is necessarily a constant, hence £ = zel? where j8 is a constant real angle. Since the unit circle is mapped on a half plane by a bilinear trans- formation, it follows that a transformation which maps a half plane into itself is necessarily a bilinear transformation. § 4-42. Normalisation of the mapping problem. Let F be the unit circle | £ | < 1 in the £-plane and suppose that a smooth region G in the z-plane can be mapped in a (1, 1) manner on the interior of F. Since F can be mapped on itself by a bilinear transformation in such a way that two prescribed linear elements correspond, it is always possible to normalise * This is often called Schwarz's lemma as another inequality is known as Schwarz's inequality. The lemma of § 4*61 is then called Schwarz's principle or continuation theorem. This second in- equality is used in § 4-81. 282 Conformal Representation the mapping so that a prescribed linear element in the region 0 corresponds to the centre of the unit circle and the direction of the positive real axis, that is to what we may call the " chief linear element." We can then, without loss of generality, imagine the axes in the z-plane to be chosen so that the origin lies in 0 on the prescribed linear element and so that this linear element is the chief linear element for the z-plane. This means that the normalised mapping function £ = / (z) satisfies the conditions/ (0) = 0, /' (0) > 0. Finally, by a suitable choice of the unit of length in the z-plane, or by a transformation of type z' = kz, we can make/' (0) = 1. The trans- formation is then fully normalised and / (z) is a completely normalised mapping function. The power series which represents the function in the neighbourhood of z = 0 is of type /(z) - z-f a2z2+ .... The coefficients a2> as> ••• *n this series are not entirely arbitrary, in fact it appears that a2 is subject to the inequality* | a2 \ < 2. To prove this we consider the function g (z) defined by the equation / (z) g (z) = 1 . We If 0<c< 1, the transformation y = g (z) maps the circular ring c < z < 1 on a region A in the y-plane bounded by a curve C and a curve Cc which can be represented parametrically by the equation cy = e-«* -f 6xc + &2cV* + c3o> (c, a), where a is the parameter and co (c, a) remains bounded as a varies between 0 and 277. Writing 62 = be2*, cd = 1, we remark that the equation ^1 V cJO gives the parametric representation of an ellipse with semi-axes d + be, d — be respectively, and so the area Ac of the curve Ce differs from nd2 by cB, where | B \ remains bounded as c -* 0. Now the area of the region A is a quantity Ac' given by the equation -l+ S n | bn I2 - S n \ bn I2 c n-2 n-2 and Ac > Ac'-, also as c -*> 0 the difference nd* — Ac' tends to the area A of the region enclosed by C. Since A > 0 we have the inequality S n\bn\*<\. n-2 Now the function [<7 (z2)]* = ^ -f ftz+..., 20!=^ likewise maps the unit circle | z | < 1 on a smooth region, and so by the last theorem , Q , \ Pi\ < !• * See, for instance, L. Bieberbach, Berlin. Sitzungsber. Bd. xxxvm, S. 940 (1916). Inequalities 283 Since bl= — a2 the last inequality becomes simply | «2 |2< 4 or | a2 | < 2. We have | & | = 1 when and only when /?2 = /?3 = ... = 0, consequently | a2 | = 2 when and only when [g (z2)]^ = - -f zeicr, where a is real, z or gr(z) = (z~* + eM)2, that is, when / (z) = ( L -*^ . EXAMPLES 1. A transformation which maps the unit circle into itself in a one-to-one manner and transforms the chief linear element into itself is necessarily the identical transformation. [Schwarz, and Poincare".] 2. A region enclosing the origin which can be mapped on itself with conservation of the chief linear element consists either of the whole plane or of the whole plane pierced at the origin. [T. Rad6, Szeged Acta, 1. 1, p. 240 (1923).] 3. If the region | z \ < 1 is mapped smoothly on a region W in the w- plane by the function w =f(z) = z + ajz2 + agZ3 -f- ..., prove that, when | z \ = r < 1, Hence show that if WQ is a point not belonging to the region W I wo I < i- The value | w0 \ = J is attained at the point w0 = Je"*" when 4. If a region W of the w-plane is mapped smoothly on the circle | z \ < 1 by the f unction w =/(z) and if Z19 Z2 are any two points which do not lie within the circle, then [G. Pick, Leipziger Berichte, Bd. LXXXJ, S. 3 (1929).] § 4-43. The derivative of a normalised mapping function. Now let / (z) be regular in the unit circle | z | < 1, which we shall call K. We shall study the behaviour of /' (z) in the neighbourhood of an interior point z0 of K. Let z0 be the conjugate of z0, then the transformation z' - Z ~ *° 1 ZZ0 maps the circle K into itself and sends the point z0 into the point z' = 0. Thus 284 Conformal Representation /*(z)=/(iz+-!°2)-/(z<>)> Writing we see that the function /* (z) maps the circle K on a smooth region and leaves the origin fixed. If z0 = reie we have by Taylor's theorem /* (z) - z (1 - r»)/' (z0) + ¥2 (1 - ^) [(1 - r2) /" fo) - 2z0/' (z0)] + - • The function! *M-<.^ta>-' + ^1+"- is thus a normalised mapping function, and so by the theorem of § 4-42, I A2 | < 2, i.e. '(1 Writing z0J v;o; - P + iQ - r ^ (w -1- iv), where/' (2) — eu+tv and ^ and v are real, we have the inequality 4r Therefore 4 - 2r : ]_ , 4 -f 2r 4 4 3v "" 1 - ra< 3r ^ 1 - r2" Integrating between 0 and r we obtain the twa inequalities J 1 - r ^ ^ , I -f r log ( 1 ~ rFhe first of these may be written in the more general form . (1+ |z|)'~ /' (z) /'(O) 1 I- (1 - where now ^ =• f (z) is a function which maps K on a smooth region not f This function was used by L. Bicberbach, Math. Zeitschr. Bd. iv, S. 295 (1919), and later by R. Nevanlinna, see Bieberbach, Math. Zeitechr. Bd. ix, 8. 161 (1921). The following analysis which is due to Nevanlinna is derived from the account in Hurwitz-Courant, Funktionentheone, S. 388 (1925). t The first of these was given by T. H. Gronwall, Comptes Rendus, t. CLXII, p. 249 (1916), and by J. Plemelj and G. Pick, Leipziger Ber. Bd. LXVIII, S. 58 (1916). In the form k(r)<\f'(z)\ <h(r) • it is known as Koebe's Verzerrungssatz (distortion theorem). The precise forms for k (r) and h (r) were derived also by G. Faber, Munchener Ber. S. 39 (1916) and L. Bieberbach, Berliner Ber. Bd. xxxvm, S. 940 (1916). The inequality satisfied by | v \ was discovered by Bieberbach, Math. Zeitschr. Bd. iv, S. 295 (1919). Properties of the Derivative 285 containing the point at infinity but is not necessarily a normalised mapping function. The second theorem is called the rotation theorem, as it indicates limits for the angle through which a small area is rotated in the conformal mapping. The other theorem gives limits for the ratio in which the area changes in size. This theorem has been much used by Koebe* in his investigations relating to the conformal representation of regions and has been used also in hydrodynamics^ and aerodynamics. When / (z) maps K on a convex region it can be shown that | /' (z) \ lies within narrower limitsj. Study has shown, moreover, that in this case any circle within K and concentric with it also maps into a convex region§. Many other inequalities relating to conformal mapping are given in a paper by J. E. Littlewood, Proc. London Math. Soc. (2), vol. xxm, p. 481 (1925). § 4-44. The mapping of a doubly carpeted circle with one interior branch point. Let P be a point within the unit circle | z' \ < 1 and let r2eld (0 < r < 1) be the value of z' at P. The transformation || (1 + r2)z- 2re* . , x ,.. z = *2£— (i^r'je*'^ (A) satisfies the conditions </> (0) = 0, </>' (0) > 0, | z' \ = \ z j, when z \ =- 1, and so represents a partially normalised transformation which maps the unit- circle in the z-plane on a doubly carpeted unit circle in the z'-plane, the two sheets having a junction along a line extending from P to the boundary. We shall regard this line as a cut in that sheet which contains the chief element corresponding to the chief element in the z-plane. It is evident that | z' \ < \ z \ whenever | z \ < 1 , and so | z \ < 1 when | z | < 1. This means that | z \ < \z\ whenever | z' \ < 1. From this we conclude that for all values of z' for which | z' \ < r2 there is a positive number q (r) greater than unity for which | z \ > q (r) \ z' \ . Indeed, if there were no such quantity g (r) there would be at least one point in the circle | z' \ < r2, for which z \ = | z' \ . An expression for q (r) may be obtained by writing 2r '-l + r" * = ?*•. and considering the points s, l/s on the real axis. If El , R2 are the distances of the point z from these points respectively we have I 'I- -at * Gott. Nachr. (1909); Crdle, Bd. cxxxvm, S. 248 (1910); Math. Ann. Bd. LXIX. f Ph. Frank and K. Lowner, Math. Zeitschr. Bd. in, S. 78 (1919). t T. H. Gronwall, Comptes Rendus, t. CLXII, p. 316 (1916). § E. Study, Konforme Abbildung einfazh zusammenhiingender Bereiche, p. 110 (Teubner, Leipzig, 1913). A simple proof depending on a use of Schwarz's inequality has been given recently by T. Rado, Math. Ann. Bd. en, S. 428 (1929). H C. Caratheodory, Math. Ann. Bd. Lxxn, S. 107 (1912). 286 Conformal Representation The oval curve for which pR1/sR2 = r2 lies entirely within a circle R2/R1 = constant which touches it at a point #0 = — p on the real axis for which / , \ 2 ps), r l + r, [(2 + 2r*)i - 1 + r«]. The constant is found to be and we may take this as our value of q (r). We can see that it is greater than 1 when r < 1 because 2 f 2r4 - (1 -f r - r2 + r3)2 - (1 - r4) (1 - r)2. It is clear from the inequality | z \ > q (r) \ z' \ that points corre- sponding to those which lie within the circle | z \ < r2 in the z' -plane lie within the larger circle | z | < r2q (r) in the z-plane. If r2 is the minimum distance from the origin of points on a closed continuous curve C" which lies entirely within the unit circle | *' | < 1, the transformation (A) maps the interior of C' into the interior of a closed* curve C which lies entirely between the two circles | z \ < 1, | z \ = r2q (r). The shortest distance "from the origin to a point of C may be greater than r2q (r) but it lies between this quantity and r, i.e. the value of | z | corresponding to the branch point z = eier2. This second minimum distance may be used as the constant of type r2 in a second transformation of type (A). Let us call it rx2 and use the symbol Cl to denote the curve into which C is mapped by the new transformation. The minimum distance from the origin of a point of this curve is a quantity r22 which is not less than a quantity r-^q (rx) associated with the number rx . If we consider the worst possible case in which the minimum distance for a curve Cn+l derived from a curve Cn with minimum distance rn2 is always rnzq (rn), we have a sequence of numbers rl9 r2, ... rn, which -are derived successively by means of the recurrence relation • W = ,-^V, [(2 + 2r.«)* - 1 + rn»]. i ~r rn » Since rn < 1 for all values of n and rn4l> rn, the sequence tends to a limit R which must be given by the equation R* = -Jj-fr [(2 + 25*)* - 1 + R*l This equation gives the value R = 1. Hence as n ->oo the curve Cn lies between two circles which ultimately coincide. * The curve C may in some cases close by crossing the line which corresponds to the cut in the z'-plane. This will not happen if the cut is drawn so that it does not intersect C' again. Sequence of Mapping Operations 287 The convergence to the limit is very slow, as may be seen by considering a few successive values of rn2 : r 2 r 2 r 2 rl r2 r3 •25 -283 -309 The best possible case from the point of view of convergence is that in which /i2 = r. This case occurs when the curve C" is shaped something like a cardioid with a cusp at P. Though useful for establishing the existence of the conformal mapping of a region on the unit circle, the present transformation is not as useful as some others for the purpose of transforming a given curve into another curve which is nearly circular, unless the given curve happens to be shaped something like a cardioid or a Iima9on with imaginary tangents at the double point. We shall not complete the proof of the existence of a mapping function for a region bounded by a Jordan curve. This is done in books on the theory of functions such as those of Bieberbach and Hurwitz-Courant. Reference may be made also to the tract on conformal transformation which is being written by Caratheodory*, to E. Goursat's Cours d' Analyse Mathematique, t. in, and to Picard's Traite d' Analyse. § 4-45. The selection theorem. Let us suppose that the set or sequence of functions u± (x, y), u2 (x, y), u3 (x, y), ... possesses the following pro- perties : (1) It is uniformly bounded. This means that in the region of definition E the functions all satisfy an inequality of type I u8 (x, y) | < M , where M is a number independent of s and of the position of the point (x, y) of the region R. (2) It is equicontinuous^ . This means that for any small positive number c there is an associated number S independent of s, x and y but depending on € in such a way that whenever (x' - xY + (yr - y)2 < S2 we have | u8 (x' ', y') - us (x, y) \ < \c. We now suppose that the sequence contains an unlimited number of func- tions and that an infinite number of these functions forming a subsequence ^wi (x> y)> Um2 (x> y}> ••• can be selected by a selection rule (m). Our aim now is to find a sequence (1), (2), (3), ... of selection rules such that the "diagonal sequence " un (x, y), u22 (x, y), ... converges uniformly in R. * Carathe"odory's proof is given in a paper in Schwarz- Festschrift, and in Math. Ann. Bd. LXXII, S. 107 (1912)., Koebe's proof will be found in his papers in Journ. ftir Math. Bd. CXLV, S. 177 (1915); Acta Math. t. XL, p. 251 (1916). f The idea of equal continuity was introduced by Aecoli, Mem. d. R. Ace. dei Lincei, t. xvm (1883). 288 Conformed Representation The first step is to construct a sequence of points Px, P2, ... everywhere dense in R. This may be done by choosing our origin outside R and using for the co-ordinates of Ps expressions of type XB = p2~", y,= p'2-"9 q>0 where p, p' and q are integers, and where the index s = / (p, p', q) is a positive integer with the following properties: 1 (P> Pr><l)> I (Po> Po'> 9o)> whenever q > q0, I (P> />'» ?) > I (Po> Po'> 7), whenever p > p0, L (P, P', (l) > l (P> Po, ?)» whenever p' > p0'. Since the functions MS (a;, y) are bounded, their values at Pl have at least one limit point Ul (xl, y^. We therefore choose the sequence uln (x, y) so that it converges at PL to this limit U1 (x1^ y^). Since, moreover, the func- tions uln (x, y) are uniformly bounded their values at P2 have at least one limit point U2(x2,y2)'., we therefore select from the infinite sequence u\n (x> y} a second infinite sequence u2n (x, y) which converges at P2 to U2 (x<2, y2)- These functions u2n(x,y) are uniformly bounded and their values at P3 have at least one limit point £/3 (x3, ?/3), we therefore select from the sequence u2n (x, y) an infinite subsequence u%n (x, y) which con- verges at P3 to C73 (x3, ?/3), and so on. We now consider the sequence uu (x, ?/), u22 (x, y), ^33 (^, y), Since the functions are all equicontinuous we have I ^m™ (*',</') - umm(x,y) | < Je for any two points P and P' whose distance PP' is not greater than S. Next let 2~q be less than 8 and let r be such that the set P19 P2, ... Pr contains all the points of R for which q has a selected value satisfying this inequality, then a number N can be chosen such that for m > N \Utt(Xs,ys) ~ ttmmfon!/*) I < !* for all values of Z greater than m and for all points (xs, ys) for which q has the selected value. This number ^V should, in fact, be chosen so that the sequences umm (x, y) converge for all these points Ps. These points P8 form a portion of a lattice of side 2~Q and so there is at least one of these points P' within a distance 8 from z. We thus have the additional in- equalities I uu(x,y) - Uu (x',yf) | < K I utt (#', y') - umm (xf, y') | < $€, KU (x, y) - umm (x, y) | < €. This inequality holds for all points P in R and proves that the diagonal sequence converges uniformly in R to a continuous limit function u (x, y). There is a similar theorem for sequences of functions of any number of variables, and for infinite sets of functions which are not denumerable. Equicontinuity In the case of a sequence of functions /x (2), f2 (z), ... of a complex variable z = x -f iy there is equicontinuity when I/. (*')-/. (3) I < 4* for any pair of values z, z' for which | z' — z \ < S, S, as before, being independent of s and of the position of z in the region E. A sufficient condition for equicontinuity, due to Arzela*, is that Z'-Z < for all functions fs (z) of the set and for all pairs of points z, z' of the domain, Ml being a number independent of s, z and z' '. We need in fact only take SJ^j = ^e to obtain the desired inequality. In the particular case when each function /s (z) possesses a derivative it is sufficient for equicontinuity that |// (z) | < M2, where M2 is inde- pendent of s and z. The result follows from the formula for the remainder in Taylor's theorem. Montel| has shown that if a family of functions fs (z) is uniformly bounded in a region R it is equicontinuous in any region R' interior to R. Suppose, in fact, that | /, (z) | < M for any point z in R and for any f unction fs (z) of the family, the suffix s being used simply a.s a distinguishing mark and not as a representative integer. Let D be a domain bounded by a simple rectifiable curve G and such that R contains D while D contains R'. We then have for any point £ within R' ,, ,w If f* (z)dz Therefor* where I is the length of C and h is the lower bound of the distance between a point of G and a point of R' . This inequality shows that the functions fs (z) are equicontinuous in R' '. Now if f8 (z) = us (x, y) -f- iv8 (x, y) where ua and vs are real, the func- tions us, vs are likewise equicontinuous and uniformly bounded in R' . We can then select from the set ua a sequence uu , w22 , u^ , . . . which converges uniformly to a function u (x, y) which is continuous in R'. Also from the associated sequence vn , v22 , v& , . . . we can select an infinite subsequence vaa, vbb, vce, ... which converges uniformly in R' to a continuous function v (x, y). The series fa* (*) + [/*> (Z) - faa (*)] + L/cc (*) ~ /» (*)] + ... then converges uniformly to u (x, y) + fy (a;, y), which we shall denote by * Mem. delta R. Ace. di Bologna (5), t. vm. t Annales de Vtfcole Normale (3), t. xxiv, p. 233 (1907). B 19 290 Conformal Representation the symbol / (z). On account of the uniform convergence the function / (z) is, by Weierstrass' theorem, an analytic function of z in J?'. Indeed if /,(n) (z) denotes the nth derivative of any function f8 (z) of our set and C' is any rectifiable simple closed curve contained within J?', we have by Cauchy's theorem and the property of uniform convergence faa(n)(z) + [fn(n)(z)~faa(n) (*)]+••• n\ Since/ (£) is continuous in J?' and on C' the integral on the right represents an analytic function. When'n = 0 this tells us that the sequence /oa («)>/» (*), — converges to an analytic function which, of course, is / (z). When n ^ 0 the relation tells us that the sequence /aa(n) (z), fbb(n) (z), ... converges to/(n) (z). We may conclude from Cauchy's expression for fs(n) (z) as a contour integral that/s(n) (z) is uniformly bounded in any region containing Rr and contained in R. It then follows that the set/s(n) (z) is equicontinuous in R . Hence from the sequence faa (z), fbb (z),fcc (z), ... we can select an infinite sequence faa (z), fftft (z),/yy (z), ... such that /««' (z),fftftr (z),/y/ (z), ... con- verges uniformly in R' to an analytic function which can be no other than /' (z). At the same time the sequence faa (z),/^ (z), ... converges uni- formly to/(z). This process may be repeated any number of times so as to give a partial sequence of functions converging uniformly to / (z) and having the property that the associated sequences of derivatives up to an assigned order n converge uniformly to the corresponding derivatives of We now consider a sequence of contours Cl) (72, ... Cn having for limit the contour CQ which bounds R, the contours Cl9 (72, ... Cn bounding domains Dl9D%9 ..., each of which contains the preceding and has .R as limit. From our set of f unctions fs (z) we can select a sequence fsl (z) which converges uniformly in Dl towards a limit function, from* the sequence /«i (z) we can cun* a new sequence f82 (z) which converges uniformly in D2 to a limit function and so on. The diagonal sequence /n (z),/22 (z), ... then converges uniformly throughout the open region .R to a limit function. Hence we have Montel's theorem that an infinite set of uniformly bounded analytic functions admits at least one continuous limit function, both boundedness and continuity being understood to refer to the open region R in which the functions are defined to be analytic. For further developments relating to this important theorem reference must be made to Montel's paper. For the case of functions of a real variable Mapping of an Open Region 291 A. Roussel* has recently invented a new method. The selection theorem has been extended by Montel to functions of bounded variation. § 4-46. Mapping of an open region. Let B be a simply connected bounded region which contains the origin 0 and has at least two boundary points. Let S be the set of analytic functions fs (z) which are uniform, regular, smooth and bounded in R and for which /s(0) = o, /;(0)=i, \f.(z)\<M. Let Us be the upper limit of | fs (z) \ in R and let p be the lower limit of all the quantities Us. There is then a sequence fr (z) of the functions/, (z) for which Ur -> p. Since, moreover, this sequence is uniformly bounded, we can apply the selection theorem and construct an infinite subsequence which converges uniformly to a limit function / (z) in any closed partial region R' or R. This function / (z) is a regular analytic function in R and satisfies the conditions/ (0) = O,/' (0) = 1. Being a uniform limit function of a sequence of smooth mapping functions it is smooth in R and its U is p. The function / (z) thus maps R on a region T which lies in the circle with centre O and radius p. If T does not completely fill the circle, there will be a value re**, with r < p, which is not assumed by our function / (z) in R. We shall now show that this is impossible and that consequently T does fill the circle. Let r = a2p, then a < 1 and if we write f (z] - 2a 0(>i* J7 (*) -JL h(Z}~ l + a*pe av(z)~- 1' r / v-,0 /(z) - tfpe* where [f (* ] --zfT N ~ ^ > L v /J a2/ (z) — pe*v v (0) = a, we have /0 (0) = 0, /</ (0) = 1 and the function /0 (z) is uniform, regular, smooth and bounded in R. Now let UQ be the upper limit of /0 (z) in R. We may find an inequality satisfied by this quantity by observing that v (z) — a av (z) — 1 is of the form ri/r2, where rly r2 are the distances of the point v (z) from the points a, I/a respectively which are inverse points with respect to the circle | z \ = 1. On the other hand, a2 | v(z) |2 = P!//^, where pl9 p2 are the distances of the point f (z) from the points a^pe**, a"2pe^ which are inverse points with respect to the circle | z \ — p. Now * Journ. de Math. (9), t. v, p. 395 (1926). See also Bull, des Sciences Math. t. LKL, p. 232 (1928). 19-2 292 Conformal Representation the point / (z) lies either on this circle or within it and so p!/p2 has a value which is constant either on | z \ = p or on a circle within | z \ = p and with the same pair of inverse points. This constant for a circle with this pair of inverse points has its greatest value for the circle | z \ — p if circles lying outside this circle are excluded. This greatest value is, moreover, ^^ - ^ p - a~2p Hence we have the inequality | v (z) |2< l. By a similar argument we conclude that rx/r2 has its greatest value when the point v (z) is at some place on the circle | z \ = 1 and this value is a. Hence v (z) — a av (z) - 1 and so U(} < p. We have thus found a function for which C70 < p, and this is incompatible with the definition of p as the lower limit of the quantities U K. The region T must then completely fill the circle of radius p and so the function/ (z) maps E on this circle. The radius p is consequently called the radius of the region R. This analysis, which is due to L. Fejcr and P. Riesz, is taken from a paper by T. Rado*. The analysis has been carried further by G. Julia| who first selects from the functions fs (z) the polynomials p^(n) (z) of degree n. Among these polynomials there is one polynomial p(n} (z) whose maximum modulus has a minimum value mn. It is clear that mn > p. Julia specifies a type of region R for which the sequence p(n) (z) possesses a limit function / (z) mapping the region R on the circle of radius p. § 4-51. Conformal representation and the Green's function. Consider in the .T?/-plane a region A which is simply connected and which contains the origin of co-ordinates. We shall assume that the boundary of A is smooth bit by bit. We write for the Green's function associated with the origin as view-point, r being short for (a:2 -f ?/2)4. Let us write H (0, 0) = log r log (!//>), then r is the capacity constant or constant of Robin J. Now let Z= <f>(z) = z -f c2z + c3z3 -|- ... be the uniquely determined function which maps the interior of a circle | Z | < p on A in such a manner that <f> (0) = 0, c£' (0) = 1. * Szeged Ada, 1. 1, p. 240 (1923). | Complex Rendus, t. CLXXXIIT, p. 10 (1926). J The boundafy of A may also be taken to be a closed Jordan curve, in which case r is the transfmite diameter. Relation to the Green's Function 293 It will be shown that the Green's function G (x, y) is 0 (x, y) = log (p/r), f=|^(2)|. Bieberbach* has proved a theorem relating to the area of the region A which is expressed by the inequality area > Tip2. This means that among all regions A for which H (0, 0) has a prescribed value the circle possesses the smallest area. For the theorem relating to the Green's function we may, with ad- vantage, adopt a more general standpoint. Let us suppose that the transformation w = / (z) maps the area A on the interior of a unit circle in the w-plane in such a way that to each point of the circle there corre- sponds only one point of the area A and vice versa. Let the centre of the circle correspond to the point z0 of the area A , then z0 is a simple root of the equation / (z0) == 0 and/ (z) = 0 has no other root in the interior of A. This is true also for the boundary if it is known that there is a (I, 1) correspondence between the points of the unit circle and the points of the boundary of A . We may therefore write 'fW-k-zJe*™, where the function p (z) is analytic in A. Putting p (z) = P + iQ, z — ZQ = re1*, where P, Q, r and 6 are all real, we have w=f(z) = exp {log r + P + i (Q + 9)}. Now, by hypothesis, the boundary of A maps into the boundary of the unit circle, therefore log r + P must be zero on the boundary of A . This means that log r -f- P is a potential function which is infinite like log r at the point (x0, ?/()), is zero on the boimdary of A and is regular inside A except at (#0, yQ). This potential has just the properties of the function G (x, y\ XQ, 3/0), where G (x, y, x0, y0) is the Green's function for the area A when (x0, y0) is taken as view-point, consequently the problem of the conformal mapping of A on the unit circle is closely related to that of finding the function G. .Writing a = Q -f- 0, 0 < a < 2-rr, we have on the boundary of the circle dw = iel*dcr, while on the boundary of A dz= \dz\ e*+, where ifj is the angle which the tangent makes with the real axis. Since dzfdw is neither zero nor infinite, the function dz\ dw) — i log ( i L. Bieberbach, Rend. Palermo, vol. xxxvm, p. 98 (1914). This theorem is discussed in § 4-91. 294 Conformal Representation is analytic within the circle and its real part takes the value ^ — a on the boundary of the circle. On the other hand, the function F (w) = - i log [- i (1 - w)2 dz/dw] is analytic within the circle and its real part takes the value iff on the boundary. If 0 is a known function of a on the boundary of A, Schwarz's formula gives , 0 where k is an arbitrary constant. The preceding formula then gives z by means of the equation ^ M ^ _ <•? — n I _ _ (O - V I -j~ ~\~f) • )W>(1 - w)* The relation between </r and a is partly known \fhen the boundary of A is made up of segments of straight lines but in the general case i/j is an unknown function of o- and the present analysis gives only a functional equation for the determination of 0. To see this we suppose that on the circumference of the circle F = i/j 4- ;</> where <f> and i/j are real, then | dz | = -J cosec2 - | da | e~^, 2i and the curvature of the boundary of A is and may be regarded as a known function of j/r, say (7 (i/r). Making use of the relation between cf> and i/j of § 3' 33 * (a) = ^ ~ 27T r ^' (a<>) iog [^ c°sec2 a° ?~] da°' where 6 is a constant, we obtain the functional equation* 4(7(0) sin^ where 0 (a) is defined by the foregoing equation. EXAMPLE Prove that [K. Lowner.] §4-61. Scliwarz's lemma. It was remarked that a Taylor expansion for / (w) in powers of w — w0 is not required for points w0 on the real axis, * T. Levi Civita, Rend. Palermo, vol. xxni, p. 33 (1907); H. Villat, Annaies de rtfcole Normale, t. xxvm, p. 284 (1911); U. Cisotti, Idromeccamca piana, Milan, p. 50 (1921). Schwarz^s Lemma 295 but when f(w) is real* for real values of w belonging to a finite interval, Schwarz has shown that it is possible to make an analytical continuation of / (w) into a region for which v is negative. Let us consider an area S bounded by a curve ACB of which the portion A B is on the line v = 0 within the interval just mentioned. Let S' be the image of S in the line v = 0 and let the value otf(w) for a point w' of S' be defined as follows. We write w = u + iv9 w' — u — iv, where u, v, f, 77 are all real. The function f (w) being now defined within the region S + S' we write g (w, £) = 1/27T* (w - 0 and consider the two integrals = 9 f = f (7 (">, Js' taken round the boundaries of S and S' Since / (w) is analytic within both S and $' we have /=/(£)> /' = 0 when £ lies within S, 7=0, /'=/(£) when £ lies within S'. Hence in either case 7 + /'=/(£) and so /(£)=( <7(t0,£)/(MOdt0, for the two integrals along the line ^4 J3 are taken in opposite directions and so cancel each other. Now the integral in this equation can be expanded in a Taylor series of ascending powers of £ - £0 for any point £0 within the area S -f S' whether £0 is on the real axis or not. The integral in fact represents a function which is analytic within the area S f S' and can be used to define /(£) within S + /S". In. this case, when £0 is on AB, f (£) can be expanded in a power series of the foregoing type and the coefficients in this series, being of type are all real. • * It is assumed here that / (w) has a definite finite real integrable value for these real values of w. In a recent paper, Bull, des Sciences Math. t. LIT, p. 289 (1928), G. Valiron has given an extension of Schwarz's lemma in which it is simply assumed that the imaginary part irj of f (w) tends uniformly to zero as v -> 0. If, then, the function / (w) is holomorphic in the semicircle | w \ < R, v > 0, it is holomorphic in the whole of the circle | w \ < R. 296 Conformal Representation Let us now use z0 to denote the value of z corresponding to this value £0 of w. The equation z - z, +-f(w) -/(U = a (w - £0) + b (w - £0)* + c (w - &)» 4- ... can be solved for w — £0 by the reversion of series if a ^ 0, and the series thus obtained is of type w-£0 = A(z-z0) + B(z- 2o)2 + C (z - z,,)3 + ... , where the coefficients A, B, C are all real. The exceptional case a — 0 occurs only when the correspondence between w and z at the point £0 ceases to be uniform. § 4-62. The mapping function for a polygon. Let us now consider an area A in the z-plane which is bounded by a contour formed of straight portions L19 L2, ... Ln. Let z0 denote a point on one of the lines L and let tin be the angle which this line makes with the real axis, also let WQ be the value of w corresponding to z. It is easily seen that the function / (w) - (z - z0) e-*- has the properties of a mapping function for points z within A, and consequently also for the corresponding region in the w-plane; it is real when the point z is on the line L in the neighbourhood of z0 and changes sign as z passes through the value z0; consequently, when considered as a function of w it is real on the real axis and changes sign as w passes through the value w0. Schwarz's lemma may, then, be applied to this function to define its continuation across the real axis and it is thus seen that we may write e-ih« (z _ 2()) =(20- WQ) P (W - W0), where P (w — WQ) denotes a power series of positive integral powers of w — WQ including a constant term which is not zero. From this equation it follows that in the neighbourhood of the point w0 where P0 (w — w0) is real when w and w0 are real. Taking logarithms and differentiating again, we see that the function = * (log &\ dw\ * dw) is real and finite in the neighbourhood of w = WQ. Next, let zl denote the point of intersection of two consecutive lines L, L' ', intersecting at an angle 0:77; the argument of zl — z varies from hrr to fnr — «TT as the point z passes from the line L to L' through the point of intersection (Fig. 19). Hence the function 1 J= [(«! - z)e~lh*]a Mapping Function for a Polygon 297 is real and positive on L and negative on L' . Moreover, it has the required properties within A , and when considered as a _, function of w it has the required mapping pro- perties in the region corresponding to A and is real on the real axis. By Schwarz's lemma we may continue this function across the real axis and may write for points w in the neighbourhood of w1 , J = (w - Wi) PI (w - Wi), where P1 (w — w±) is a power series with real coefficients and with a constant term which is not zero. This equation gives z — zl = e*hir (w — Wi)a P2 (w — w^, where P2 (w — w±) is another power series with real coefficients. This equation indicates that for points in the neighbourhood of wl dZ .. . x 1 T> / x ^ = «*•<«- «*>•-* pa(»-«a where P3 (w — w±) is a power series with real coefficients. Taking logarithms and differentiating we find that n / x d /, dz\ a — 1 m , . F (w) = j- ( log j - ) = h T (w ~ WJ, dw \ ° dw) w — w^ where T (w — wj is a power series with real coefficients. The function F ( - a~ l w — Wi is thus analytic in the neighbourhood of w — wl . For a point z2 on the boundary of A which corresponds to ^v we have (if z2 is not a corner of the polygon) 2 w w2 Therefore -T- = lp\~ dw w2 ^ \w. d A dz\ 2 1 /IN = j- ( log 3- ) = ---- h — s P! I - , d?/; \ & dw;/ te; w;2 / \w;/ where p (l/w) is a power series. The expansion for z — z2 may, indeed, be obtained by mapping the half plane w into itself by means of the sub- stitution w — — \IW-L , and by then using the result already obtained for an ordinary point z0 on L. The function F (w) is real for all real values of w, as the foregoing investigation shows, is analytic in the whole of the upper half of the w-plane and is real on the real axis, the fact that it is analytic being a 298 Conformal Representation consequence of the supposition that the inverse function z — g (w) is analytic in the upper half of the w-plane. Applying Schwarz's lemma we may continue this function F (w) across the real axis and define it analytically within the whole of the w-plane, the points on the real axis which are poles of F (w) being excluded. When | w \ is large | F (w) \ is negligibly small, as is seen from the expansion in powers of ljw\ moreover, F (w) has only simple poles corre- sponding to the vertices of the polygon A and these are finite in number. Hence, since F (w) outside these poles is a uniform analytic function for the whole w-plane, it must be a rational function. Let a, 6, c, ... 1 be the values of w corresponding to the vertices of the polygon and let arr, /?TT, ... XTT be the interior angles at these vertices, then ™ / x ^ a — I d , , dz F (w) = 2 = -=- log -,— , w — a aw & aw and there is a condition S (a - 1) = - 2, which must be introduced because t}ie sum of the interior angles of a closed polygon with n vertices is equal to (n — 2) TT. Integrating the differential equation for z we obtain z = c( (w - a)"-1 (w - &V3-1 ... (w - I)*"1 dw + C", where C and C" are Arbitrary constants. By displacing the area A without changing its form or size but perhaps changing its orientation we can reduce the equation to the form = K \(w- a)*-1 (w - by-* ...(w- I}*-1 dw, where K is a constant. This is the celebrated formula of Schwarz and Christoffel* If one of the angular points with interior angle fin corresponds to an infinite value of w, the number of factors in the integrand is n — 1 instead of n, and the equation S (a - 1) = - 2 may be written in the form S (a - 1) = - 1 - p, where now the summation extends to the n — 1 values of a which appear in the integral. Since we can choose arbitrarily the values of w corresponding to three vertices of the polygon, there are still n — 3 constants besides C and C' to be determined when the polygon is given. In the case of the triangle there is no difficulty. We can choose a, 6 and c arbitrarily; a, ]8 and y are known from the angles of the triangle and by varying K we can change the size of the triangle until the desired size is obtained. * E. B. Christoffel, Annali di Mat. (2), 1. 1, p. 95 (1867); t. iv, p. 1 (1871); Get. Werke, Bd. I, S. 265. H. A. Schwarz, Journ.filr Math. Bd. LXX, S. 105 (1869); Ges. Abh. Bd. n, S. 65. Mapping of a Triangle 299 An interesting example of the conformal representation of a triangle with one corner at infinity is furnished by the equation z — z0 = \ f (>s) ds, where / (6*) = (2a/7r) (1 — s2)%/s, w = u ~\- iv. When w lies between 1 and oo we have = ft -f ic, say, where ft is a constant and c varies from 0 to oo. Thus, the portion w > 1 of the real axis corresponds to a line parallel to the axis of y. Again, if 0 < w < 1, we may write i - ft - d, where d varies from 0 to 06. The portion 0 < w < 1 of the real axis corre- sponds, then, to a line parallel to the axis of x and extending from z --= z(i -f- ft to — oo. When — 1 < w < 0 we may write r - 1 no *-*o= f(*)ds+ f(s)ds Ji J -i = ft' 4- d', and so the corresponding line in the z-plane extends from — oo to ft' H- z0 and is parallel to the axis of x. When — oo < w < 1, we have f ~l (~l - ZD = / (*) ds- \ f (s) ds Ji Jw - ft' - ic', where c' ranges from 0 to oo, and so this part of the ^-axis corresponds to a line from ft' parallel to the i/-axis. The two lines parallel to the i/-axis can be shown to be portions of the same line separated by a gap. We have in fact b-bf=ff (s) ds - f "V (s) ds = f1 / (s) ds, Ji Ji J -I where the integral is taken along the semicircle with the points — 1, + 1 as extremities of a diameter. On this semicircle we may put andso f° i (7r/2a) (ft - ft') = i\ -*• d0 [- 2i sin 0 e^ J re fir / /3 /D\ -= i (1 - i) (cos - + i sin - J (sin 6)4 rf0 - 2i f" cos ? (sin 9)% d0 = 2i^2T (J) T (f ) = i. 300 Conformal Representation The figure in the z-plane is thus of the type shown in Fig. 20. To solve an electrical problem with the aid of this transformation we put d> = ie^x, where % is the complex potential $ -f ifi. This transformation maps the half w-plane for which v > 0 on a strip of the ^-plane lying between the lines <f> — il: 77. Performing the integration we find that log ?-±- - 2 V2 - 2 log where Fig. 20. -C, 1-1 Fig. 21. When the real part of w is large and negative the chief part of the expression for z is a log (r - 1) = - log (1 + & + ... - 1) = log 2 - - 77 77 This gives a field that is approximately uniform. On the other hand, when the real part of w is large and positive, the chief part of the expression for z is and we may thus get an idea of the nature of the field at a point outside the gap and at some distance from its surfaces. These results are of some interest in the theory of the dynamo. Another interesting example, in which the polygon is originally of the form shown in Fig. 21, gives edge corrections for condensers*. Assigning values of w to the corners in the manner indicated, the transformation is of type \ 1 [(w + ^ (w - b) dw CaC^Ce)-* V ---- -'-+-- ' J(w — 0,1)* (w - n z - G -- ---- — (w -f a2) * r. . [(q - w) ... (c6 + * J. J. Thomson, Recent Researches in Electricity and Magnetism, 1893; Maxwell, Electricity and Magnetism, French translation by Potier, ch. n, Appendix; J. G. Coffin, Proc. Amer. Acad. of Arts and Sciences, vol. xxxix, p. 415 (1903). Edge Correction for a Condenser 301 Making a8 -> 0, c8 -> oo we finally obtain where G and 6 are constants to be determined. Integrating, we find that z = C [w + I (w - 6)2 - 6 log (- w) + F], where F is a constant of integration. Since z = 0 when w? = — 1, we have F= 1- |(1 + 6)2. When ^ is small and positive, the imaginary part of z must be ^7?, and the real part must be negative. Since the argument of — w in both con- ditions are satisfied by taking C = — h/bir, therefore _ h Assuming that the potential </> is zero on A A and equal to V on BB1 , we may write ^ = 0 — i<^ = (log WJ — ITT). 7T Fig. 22. The charge per unit length on BB' from the edge (w = 6) to a point P (?# = s) so far from B that the surface density is uniform is 1 V q== ~~ ^p ~ ^ = ~ log (<S/6)l 47T Now when 5 is very small and positive, z = x -f i&, and so x + iA = _ --- (i - 26 - 2i&7r - 26 log 5). Therefore log (s/b) = irx/h + 1/26 - 1 - log 6, F TTX 1 - 26 , and so 8s= When b = 1 we have the well-known result , /] _log6j. in which it must be remembered that x is negative. 302 Conformal Representation When w is very small and negative, z — x, and so 2 Io V P W>, LTT V h ,rl , , V (h \ VV hen b — \ q — — . -. I x } , 47T//, \277 / A f/ (h \ 6-00 r/ = - — - - x }. A~L \7T ] Since ?^ -- 6 at the point B, the value of 2 for this point is 3 = - 5f- [1 - 62 - 26 log 6 - 26i7r], 2u7T and so the upper plate projects a distance d beyond the lower one, where Many important electrostatic problems relating to condensers are solved by means of conforinal representation in an admirable paper by A. E. H. Love*. The problem of the parallel plate condenser is treated for planes of unequal breadth and for planes of equal breadth arranged asymmetrically. The formulae involve elliptic functions. The hydrodynamical problems relating to two parallel planes, when the motion is discontinuous, are treated in a paper by E. G. C. Poolef. Some applications of conformal representation to problems relating to gratings are given in a paper by H. W. Richmond^. The general problem of the conformal mapping of a plane with two rectilinear or two circular slits has been discussed recently by J. Hodgkinson and E. G. C. Poole§. § 4-63. The mapping function for a rectangle. When n — 4 and a ^ p ^ y -= 8 = ^, the polygon is a rectangle and z is represented by an elliptic integral which can be reduced to the normal form z = H f dt [(1 - t*) (1 - &2*2)]-i Jo by a transformation of type w(Ct + D) - At + B. ...... (A) If, in fact, the integral is dw [(w —p) (w — q) (w — r) (w — s)]~%9 we have (Ct -f D) (w - p) - (A - Cp) t -f B - Dp, (Ct -h D2) dw - (AD - BC) dt, * Proc. London Math. Soc. (2), vol. xxn, p. 337 (1924). f ^id. p. 425. J Ibid. p. 389. § Ibid. vol. xxra, p. 396 (1925). Mapping of a Rectangle 303 and so the transformation reduces the integral to the normal form if A - Cp = B - Dp, A - Cq = Dq - B, A - Cr = k (B - Dr), A - Gs = Ic (Ds - B). These equations give C (q - p) = 2B - D (p 4- g), C (s - r) = 2kB - kD (r + s), 2A - (7 (p + <?) = JQ (q - p), 2A - C (r + s) = kD (s - r), C[s-r- k(q-p)]= kD[p + q-r- s], C [r + s - (p + q)] - D [q - p + k (r - a)], £2 (? _ p) (r _ 3) + k [(q _ p)2 + (r _ 0)3 _ (p + ? __ f __ j)S] + (? ~ P) (r ~ s) = 0. This equation gives two values of k which are both real if [(? ~ P? + (r ~ «)2 - (p + ? - r - «s)2]2 > 4 (gr - ^))2 (r - s)*, that is, if [(? - P + r ~ s)2 ~ (P + ? ~ r - 5)21 [(? ~ P ~ r + <*)2 ~ (p + ? - r - 5)2] > 0, or, if 4 (# — «s) (r — p) (q — r) (s — p) > 0. Itr^p^>q^s this is evidently true and since the product of the two values of k is unity, we may conclude that one value of k is greater than 1 , the other less than 1. This latter value should be chosen for the transfor- mation. With this value consequently, the transformation (A) transforms the upper half of the w-plane into the upper half of the £-plane. When the normal form of the integral is used the lengths of the sides of the rectangle are a and b respectively, where a - H dt [(1 - t2) (1 - W)]-* - 2HK, J-i ri/fc 6 = ff I dt [(1 - t*) (1 - 4V)]r* = HK', and where 4X and 2iK are the periods of the elliptic function sn u defined by the equation x = sn u, where u^ \X dt[(l - t*) (1 - tV)]-». Jo With the aid of this function £ can be expressed in the form t - sn (z/H). 304 Conformal Representation The modulus k may be calculated with the aid of Jacobi's well-known r(\ -fg'Ml + ) (1 f. formula in which </ = exp [— TrK'/K] = exp [— 2?r6/a]. When the region is of the type shown in Fig. 23 the internal angles of the polygon are 877/2 at four corners and ?r/2 at the other eight. The transformation is thus of the type 2 - A [(w - cj (w - c2) (w - c3) (w - c4)]* [(w - fa) ... (w - pB)]~*dw + B. A particular transformation of this type is obtained by assigning positive values of w to corners of the polygon which lie above the axis of x and negative values of w to corners which lie below the axis of x, points which are images of each other in the axis of x being given parameters whose sum is zero. The transformation is now z = Fig. 23. Fig. 24. Making the parameters ax, a2 tend to zero and the parameters 6lf 62 tend to infinity, the transformation becomes , (w* - B, and the interior of the polygon becomes a region which extends to infinity. To use this transformation for the solution of an electrical problem in which the two pole pieces in Fig. 24 are maintained at different poten- tials, we write* «, = ;CM*, x = 0 + ^, so as to map the half of the w-plane for which v > 0 on the strip — TT < <f) < TT. This will make w> = 0 correspond to z = 0 if B = 0, and the lower limit of the integral is i \/c. Writing ck = 1 we find that the lengths a and 6 in the figure are given by the equations re ^ 26= <7c ~a/(a), ./i 5 ^ f1 d$ « , . ~ [~lds j. . . - 2*a = Cc -2/ (5) - Cc -2 / (a), Jt*/c 5 JiJc s * Riemann-Weber, Differentialgleichungen der Physik, Bd. n, S. 304. Region outside a Polygon 305 where f (s) = [(1 — s2) (1 — k2s2)]*. These integrals are easily reduced to standard forms of elliptic integrals, thus cds - c ds ds ** c ds Now if we put ksr = 1, the last integral becomes ~ _ r (L~JL?!L^ ~ Ji ~7F) and we eventually find that 6 = Cc[2Ef- (I- k*)K'], a- 2Cc[2E - (1 - ia) JL]. 26 2^' - (1 - A8) #' Therefore a 2E- (I - P) K ' EXAMPLE Prove that if OABC is the rectangle with sides x = 0, x = K, y = 0, y = K' and ^ -f it/t = log (an z), we have ^ = 0 on OA, ABt BC; 0 == rr/2 on CO. Prove also that if ^ 4- itj, = log (en z), where (en z)2 4- (sn z)2 = 1, we have 0 = 0 on 0.4, 0(7; t/i = — n/2 on J5^4, J5C. See GreenhilTs Elliptic Functions, ch. ix. § 4-64. Conformal mapping of the region outside a polygon. In order to map the region outside a polygon on the upper half of the w-plane, we may proceed in much the same way as before, but we must now use the external angles of the polygon and must consider the point in the w-plane which corresponds to points at infinity in the z-plane. Let us suppose that the w-plane is chosen so that this point is given by w = i, then there should be an equation of the form — ~— C C (w - i) where the coefficients Cm are constants. This gives _? — _ i Q i dw (w — i)2 l *"' d , dz 2 -~ log ~- = . -h P (w — ^), aw ° aw w — i where P (w — i) is a power series in w — i. Since -3— log -j— is to be real it must be of the form dw & dw — 1 — - S a-^~l - 2 2 dw ° dw w — a w — i w + i% B 20 306 Conformal Representation Therefore w- a)"-1 (w - 6)*-1 ... (w - iy~l (1 4- w*)~*dw + C', ...(I) where C and C' are arbitrary constants of integration. The relation between the indices a is now S (a - 1) = 2, for the sum of the exterior angles of a polygon with n vertices is (n 4- 2) TT. The region outside a polygon can be mapped on the exterior of a unit circle with the aid of a transformation of type - a)*-1 (w - b)?-1 ... w~2dw, | a | = | 6 | = ... = 1, where, as before, S (a - 1) - 2. When the integrand is expanded in ascending powers of w~l there will be a term of type w1 which will, on integration, give rise to a logarithmic term unless the condition Sa(a- 1) = 0 is satisfied. When the polygon has only two vertices and reduces to a rectilinear cut of finite length in the z-plane, we have a = ft = 2. The second condition may be satisfied by assigning the values w — ± 1 to the ends of the cut. The transformation is now = H f (w2- l)w-2dw, and the length of the cut evidently depends on the value of H. Taking H — £ for simplicity, the transformation becomes 2z = w 4- w~l. This is the transformation discussed in § 4-73. The general theorem (I) indicates that the regibn outside a straight cut may be mapped on the upper half of the w-plane by means of the transformation ft f 1 - w2 , 2w ty O I fJIH — Z — ^ I 71i , oTo ™w — -i ," o« J (1 4- w2)2 1 4- w2 The region outside a cut in the form of a circular arc may be obtained from the region outside a straight cut by inversion. If the arc is taken to be that part of the circle z = — ie*ie, for which — a < 6 < a, the trans- formation 4-14- 2iw tan a z = — w2 4- 1 — 2iw tan a maps the region outside the arc on a half plane. Suppose that in the w-pjane there is an electric charge at the point w = i (sec a 4- tan a) = is, say, and that the real axis is a conductor. Semicircular Arc 307 The corresponding charge in the z-plane will be at infinity and the circular arc will be a conductor which must be charged with a charge of the same amount but of opposite sign. The solution of the potential problem in the w-plane is evidently i , - 1 i w — is y = 0 -f ii/f = log --- — . * V -T- "T 6 w _j_ ls mu- • i.i_ /i \ - 1 + &~x sin This gives ur = - is coth and finally X= — log ^ cosec a {z 4- 1 4- (z2 4- 2iz cos 2a — 1)1} . The two-valued function (z2 + 2^z cos 2a — l)i may be regarded as one-valued in the region outside the cut and must be defined so that it is equal to i when z = 0 and is of the form — z — i cos 2a when j z \ is very large. Changing the signs of <f> and x we have x = K [2 sin a cosh <f> sin 0 -f sin2 a sin 20], ^ = _ jf [l -f- 2 sin a cosh </> cos 0 4- sin2 a cos 20], where K~l = 1 4- 2e~^ sin a cos 0 -f e"2^ sin2 a. With the aid of these equations Bickley has drawn the equipotentials and lines of force for the case of a semicircular arc. The charge resides for the most part on the outer face, the surface density becoming infinite at the edges. The field appears to be approximately uniform ori the axis just above the centre of the circle. The field at a great distance from the circular arc is roughly that due to an equal charge at the point z = — i cos2 «, for when x is large, the equation . 1 -f e* sin a . r , . ._ _ , z = — i -.- ------ --. — = — i [1 -f ex sin a] [1 — e~* sin a] ... 1 4- e~x sin a L J may be written in the form z 4- i cosa a = — iex sin a 4- negligible terms. This point, which may be called the " centre of charge," is the middle point of that portion of the central radius cut off by the chord and the arc. On the circular arc X = i0 and z == — ie2ld. Therefore sin (0 — 0) sin a = sin 9. The surface density is thus proportional to S cos 8 4- 1 on the convex face and to S cos 6 — 1 on the concave face, S denoting the quantity S = (sin2 a - sin2 0)"*. If E is the charge per unit length of a cylindrical conductor whose 308 Conformal Representation cross-section is the circular arc and d is the diameter of the circle, the surface density a is given by Love's formula 2-rrcrd = E | sec v \ (cosec a — cos v), where sin v = — tan 0 cot a. The solution of the electrical problem of a conducting plate under the influence of a line charge parallel to the plate but not in its plane may be derived from the preceding analysis by inversion from a point 0 on the unoccupied part of the circle. Let AB be the cross-section of the plate, /)'the foot of the perpendicular from O on AB, OC1 the bisector of the angle AOB, then the surface density cr is given by Love's formula _ E OD cosec a — cos v ° = 277 OP2 "J cos ~v~\ ' where now sin v = cot a tan (P'OC'), cos v = ± cosec a — ^-p — - , A'P'C'B' being perpendicular to OC1 (Fig. 25). Thus __ E OD OA' =f (A'P' . J5'P')_* 07 ~ 277 OP2 (A'P'.WP')* O Fig. 25. This is easily converted into the expression given in § 3-81. The region outside a rectangle may be mapped on the interior of the unit circle in the £-plane with the aid of the transformation z= f ds ( 1 - 2s2 cos 2a + Ji while a transformation which maps the region outside the rectangle into the region outside the circle is obtained by using a minus sign in front of the integral. Let us use this transformation to determine the drag on a long thin rod of rectangular section which is moved slowly parallel to its length through a viscous liquid contained in a wide pipe of nearly circular section. We write log £ = iw = i (u -f iv), where v is the velocity at any point in the z-plane, then = 2* f (cos 2a - cos 2s)$ ds = 2 f (sin2 a - sin2 <$)* ds. Jo Jo Hydrodynamical Applications 309 Putting sin s = sin a sin 0, this becomes - 2 f «?^lizJE^!)_f - 2 I* (1 - sin* « si Jo (1 - sin2 a sin2 j3)* Jo - 2 f cos2 a (1 - sin2 a sin2 j8)~i Jo At the corner A immediately to the right of the origin 0 in the z-plane, we have 6 = |TT, and XA = 2E (k) - 2k'2 K (k), where k = sin a and E (k), K (k) are the complete elliptic integrals to modulus k. The drag on the half side OA of the rectangle is proportional to WA) and since sin WA — sin a sin (|TT) = sin a, we have WA = a. The drag on the side OA is thus equal to (a/2?r) times the drag of the whole rectangle. [C. H. Lees, Proc. Roy*. Soc. A, vol. xcn, p. 144 (1916).] EXAMPLE A line charge Q at the origin is partly shielded by a cylindrical shell of no radial thickness having the line charge for its axis, the trace of the shell on the #y-plane being that part of the circular arc z = ae2lB for which — TT < — 2co < 20 < 2o> < TT. Prove that the potential <f> is given by the formula (z ~ a) cos2 "> -f- (z -f a) sin2 a> -f R a) 8n „ _ (z _) where J2 denotes that branch of the radical [z2 — 2az cos 2aj -f a2]*, whose real part is positive when the point z is external to the circle. The surface density a of the induced charge at a point 0 on the charged arc is a = _ Q {Sec a> (tan2 o> - tan2 0)~* ± l}/47ra, the upper sign corresponding to the density on the concave side, the lower sign to the density on the convex side. The latter is zero when 2o> = -n-, that is, when the circle closes. [Chester Snow, Scientific Papers of the Bureau of Standards, No. 642 (1926).] § 4-71. Applications of conformal representation in hydrodynamics. Consider the two-dimensional flow round an airplane wing whose span is so great that the hypothesis of two-dimensional flow is useful. Let u, v be the component velocities, p the pressure, p the density, L the lift per unit length of span, D the drag per unit length and M the moment about the origin of co-ordinates, this moment being also per unit length. These quantities may be calculated from the flux of momentum across a very large contour C which completely surrounds the aerofoil. In fact, if /, ra are the direction cosines of the normal to the element ds, we have L -f iD = — p \ (v -f iu) (ul -f vm) ds— \ p (m -f il) ds, M = — p \ (xv — yu) (ul + vm) ds — \ p (xm — yl) ds\ 310 Conformal Representation the sign of M is such that a diving couple is regarded as positive. The equations may be rewritten in the form L + iD = l/o j (v -f fw)2 dfe - I [p + Jp (^2 -f v2)] (m -f fZ) ds, M = \p\ [(u2 - v2) (mx 4- ly) -f 2tw (my - Zx)] efo f (ly ~ mx)[p -f where 2 = x -f- iy. Now when the motion is irrotational outside the aerofoil the quantity p -f \p (u2 -f v2) is constant, also (m -f iZ) rfs = 0, (ly — mx) ds = 0, hence L + iZ> = lp \ (v 4- ft*)2 rfz. Taking the contour to be a circle of radius r, we have (y2 - x2)] ds/r f [^2 - ^2 - 2fwv] [x2 - y2 + 2ixy] ds/ir (^ -h iu)2zdz, where the symbol R is used to denote the real part of the expression which follows it. These are the formulae o^Blasius* but the analysis is merely a development of that given by Kutta and Joukowsky. The integrals may be evaluated with the aid of Cauchy's theory of residues by expanding v -f iu in the form When the region outside the aerofoil is mapped on the region outside the circle | z' \ — a by a transformation of type the flow round the aerofoil may be made to correspond to a flow round the circle by using the same complex potential x in each case. Now for our flow round the circle we may write ®X '- /v* tf 2/?'2 4- IK e a/z + Zeits.f. Math. u. Phys. Bd. Lvm, S. 90 (1909), Bd. LIX, S. 43 (1910). Region outside an Aerofoil 311 where V, a and K are constants, therefore . dv . dv dz' v + iu = i -~ = i -=£ . — - dz dz dz 'e-*- We i/c KCa inc* U' . - -f s- --- ~ ~ 27T2 If 2' = aelf* is the point of stagnation on the circle which maps into the trailing edge of the aerofoil, we have K = 2-jraU' sin (a - /?), U = nCTe—, L + iZ> - KpU'ne-* = KpU = 2<napUUr sin (a - /?). § 4-72. jT/^e mapping of a wing profile, on a nearly circular curve. For the study of the flow of an inviscid incompressible fluid round an aerofoil of infinite span, it is \iseful to find a transformation which will map the region outside the aerofoil on the region outside a curve which is nearly circular. If the profile has a sharp point at the trailing edge at which the tangents to the upper and lower parts of the curve meet at an angle a, it is convenient to make use of a transformation of type Z + KC where cr = (2 — K) TT. If a circle is drawn through the point — c in the £-plane so that it just encloses the point c, cutting the line (— c, c) in a point c -f d, say, where d is small, this circle will be mapped by the transformation into a wing- shaped curve in the z-plane. This curve closely surrounds the lune formed from two circular arcs meeting at an angle a at each of their points of junction, z == *c, z = — KC. The curve actually passes through the point — KC and has the same tangents there as the lune derived from a circle in the £-plane which passes through the points (— c, c) and touches the former circle at the point — c. If we start with the profile in the z-plane and wish to derive from it a nearly circular curve with the aid of a transformation of this type, the rule is to place the point — KC at the trailing edge and the point KC inside the contour very close to the place where the curvature is greatest*. This rule works well for thin aerofoils, but it has been found by experience that by increasing the magnitude of d an aerofoil with a thick head may be obtained from a circle, and that the thickness of the middle portion of the aerofoil is governed partly by the value of cr. Hence in endeavouring to * F. Hohndorf, Zeits. f. ang. Math. u. Mech. Bd. vi, S. 265 (1926). 312 Conformed Representation map a thick aerofoil on a nearly circular curve the point KC may be taken at an appreciable distance from the boundary. Another point to be noticed is that when a circle is transformed into an aerofoil by means of the transformation (A) the smaller the distance of the centre from the line (— c, c) the smaller is the camber of the corresponding aerofoil and the more symmetric is the head. The point KC, moreover, lies very nearly on the line of symmetry. The actual transformation may be carried out graphically with the aid of two corresponding systems of circles indicated by the use of bipolar co-ordinates. The circles in one plane are the two mutually orthogonal coaxial systems having the points (— c, c) as common points and limiting points respectively; the corresponding circles in the other plane for two mutually orthogonal systems having the points (— c, c) as common points and limiting points respectively. This is the method recommended by K£rm&n and Trefftz. Another construction recommended by Hohndorf depends upon the substitutions by which the transformation may be written in the form where The plan is to first consider the transformation from z to £, given by the equation ,1 C, ~~" C (Z ~~~ KC\ * r = t* or f— = ( — — ) . £ -f- c \z 4- KC) • This transformation may be performed graphically* by writing z0 = z + KC, £0 = J + c, when the relation becomes When $ has been found its ^th root may be determined graphically and when this is multiplied by $ the value of r is obtained, and from this £ is easily derived. Hohndorf gives a table of values of rj corresponding to different angles a. When a -= 4°, 77 = 89, when a = 8°, 77 = 44, and when a = 10°, * The transformation may also be performed graphically by writing it in the form •-5('*?) when it is desired to pass from a figure in the (-plane to a corresponding figure in the 2-plane. Aerofoil of Small Thickness 313 rj = 85. When the transformation (A) is expressed by means of series, the results are _, /c2 - 1 c2 (/c4 - 5*2 + 4) c4 + •••> r l-/c2c2 (4/c4- 5*2+ l)c4 r — ~ i __ __ v __ _ __ _ _ _i _ _i_ ^ ^ 3 z 45z3 ^"" § 4-73. Aerofoil of small thickness*. We have seen that the trans- formation z' = z + a2/z maps the circle (7 given by | z \ = a into a flat plate P' extending from z' = 2a to z' = — 2a and back. On the other hand, if the A's are small quantities the transformation £ = s{l+ S ^n(a/2)"} n-O maps C into a curve F differing slightly from a circle, and if we then put F maps into a curve II' differing slightly from a flat plate. Now for a point on F £ = a(l + r)eie, where 6 is a real angle and r a real quantity which is small ; therefore to the first order in r £ = 2a (cos 0 + ir sin 0), and so £' = 2a cos 0, rjf = 2ar sin 0. For points on (7 and P' we may use a real angle co and write z = aeia>, a?' = 2a cos co, y' == 0, then (1 4- r) e«»--) = 1 + S ^«e-'»-, n»0 i and since 0 — o> is small we have to the first order, with An = Bn + iCn , r = S (J5n cos ^^6o -f <7n sin 7^0), 0 — </> = 2 (Cn cos nco — -Bn sin /io>). Hence by Fourier's theorem D r AJQ (* ^ c°s n® ^ 7r.Bn = r cos n0d6 = -±- —. — w dO, * H. Jeffreys, Proc. Roy. Soc. A, vol. cxxi, p. 22 (1928). 314 Conformal Representation Since sin 0 and sin n0 are odd functions of 0, whilst cos nO is an *even function of 6, it appears that Cn depends on the sums and Bn on the differences of the values of rj' corresponding to angles ± #; thus the (7n's depend on the camber of the aerofoil, the J?n's on its thickness. When 6 is small, that is, for points near the trailing edge of the aerofoil, we have approximately v) _ r sin 0 __ 2r 2a~~-~? ^ r~"cos 0 = ~0 ' and when TT — 0 is a small quantity co we have Thus r vanishes at 0 = 0 because the slope of the section is finite there ; but at 0 = TT the section and the axis meet at right angles at a point which may be called the leading edge. If the curvature at this point is l/R we have to a close approximation nr» V2 4a2r2sin2# _ . ., m . 9 2jR = ,.— -s- = :r~ ,- --- -„-, = 2ar2 (1 — cos 5) = 4ar2, ^ + 2a 2a (1 4- cos 0) v ; consequently r is finite and equal to (R/2a)% at the leading edge. If at a great distance from the circle C the flow in the z-plane is represented approximately by a velocity U making an angle a with the axis of x, we have IK x = </, + ie/r = C7ze-* -f Ue^at/z + ^ log (z/a), where K is the circulation round the cylinder. Taking x to be the complex potential for a corresponding flow in the £'-plane the component velocities (u9 v) in this plane are given by the equation To determine K we make the velocity finite at the trailing edge where f ' is a maximum and 0 = 0, r = 0, £ = a, -^ r = 0. Hence d^/dz is zero and so But when 0=0 say, where j8 == S Cn . Therefore /c = ±naU sin (a + Thin Aerofoil 315 Now {1-2 '(n - 1MM (a/zf«}"(l - a'/T2) « 1 ' (1 + fl> + 41o/g')/2"£' , " + IK Therefore by the Kutta-Joukowsky theorem the lift per unit span of the aerofoil is L - - = 477/>a (1 + 50) sin (a + jB) F2, 7(1 + J50) = J7, 1 H~ -^0 and the lift coefficient is #L = (L/4paV*) = 77 (1 + J?0) sin (« + )8). The thickness thus affects the lift through J30, which is a positive constant for a given wing. The moment about 0, that is a point midway between the leading and trailing edges of the aerofoil, is equal to np times the real part of the coefficient of -f i£'~2 in the expansion of (-=£ j . This coefficient is and so to the first order in the J3's and C"s the moment is M , where M - Z-n-pVW {C2 cos 2a - (1 + B2) sin 2a f 2Bl cos a sin (a + ]8) 4- 2^ sin a sin (a + ]3)}. The moment about the leading edge is where terms of orders a2, aBn> aCn have been retained, but terms of orders a3, a2J?n, a2Cn, dropped. When squares and products of the JB's and C"s are neglected, the moment coefficient KM is = \KL (1 + B. + B,- The moment coefficient at zero lift is thus and is independent of the thickness to this order of approximation. The thickness, however, affects the coefficient of KL. 316 Conformal Representation For further applications of conformal representation in hydrodynamics the reader is referred to H. Glauert's Aerofoil and Airscrew Theory (Cam- bridge, 1926) and to H. Villat's Lemons sur VHydrodynamique (Gauthier- Villars, Paris, 1929). § 4« 81. Orthogonal polynomials associated with a given curve*. Let/ (z) be a function which is defined for points of the z-plane which lie on a closed continuous rectifiable curve C which is free from double points. If ds = I dz I , the integral r f(z)ds J C denotes as usual the limiting value lim 2 m->co f»»l where z0 , zl , z2-5 . . . represent successive points on C for which lim [Maximum value of | zv — zv_\ \ for 0 < v < m] — 0, (zl = zm), in — >• oo and £„ denotes an arbitrary point of C which lies between zv^ and zv. We have in particular /• ds = l, Jc where I denotes the length of the curve C. We shall suppose now that the unit of length is chosen so that I = 1 . Using z to denote the complex quantity conjugate to z, we write D0 =sAoo = n. Let Hmn denote the co-factor of hmn in the determinant Dn , and let an be a constant whose value will be determined later ; then, if Pn (z) = an (H0n + zHln + ... zn Hnn), it is easily seen that f Pn (z) zvds - an (hQvH0n + hlt,Hln -f ... hn¥Hnn) Jc - an Dn for v - n, Pn (z) |2 dz = ananHnnDn - an * See a remarkable paper by Szego, Jtfa^. Zette. Bd. rx, S. 218 (1921). Orthogonal Polynomials 317 The polynomials thus form an orthogonal system which is normalised by choosing an, so that an = an = (A» Ai-i)"*- If Pm (z) denotes the complex quantity conjugate to Pm (z), the ortho- gonal relations may be written in the form Pn (z) Pm (z) ds= 0, m*n Jc = 1, m = n. Let us now suppose that C is an analytic curve and that £ = y (z) is the function which maps the interior of C smoothly on the region | £ | < 1 of the £-plane in such a way that y (a) = 0, y (a) > 0. Since C is an analytic curve y (z) is also regular and smooth in a region enclosing the curve C. It is known, moreover, that there is one and only one function, z = g (£), which is regular and smooth for | £ | < 1 and maps the interior of | £ | = 1 on the interior of the curve C. The derivatives of the functions y (z), g (£) are connected by the relation y (z) g' (£) = 1, where z and £ are associated points of the two planes. Our aim now is to show that o rz = lim ~-,J-r [Kn(a,w)]*dw, n_>oo J^n Va> a) J a where Kn (a, z) = P0 (a) P0 (z) + ... Pn (a) Pn (z). We shall first of all prove an important property of the polynomial Kn (a, z). Let a be arbitrary and Gn (z) a polynomial of the nih degree with the property f \Gn(z)\*ds = 1, Jc then the maximum value of | Gn (a) \ 2 is Kn (a, a), and this value is attained when Gn (z) = eKn (a, z) [Jf n (a, a)]~~i, where e is an arbitrary constant such that | e | = 1. Let us write Gn (z) = ^0P0 (z) + ^P! (z) + ... tnPn (z), where the coefficient tv is determined by Fourier's rule and is = f J We then have (z)\*ds= |*0|*-f |^|2+ ... \tn\\ = f J C G (a) = /0P0 (a) + ^P! (a) + ... ^nPn (a), and by Schwarz's inequality 318 Conformal Representation The sign of equality can be used when t, = e!\W[Kn(a,a)]-*-, that is, when On (z) = eKn (a, z) [Kn (a, a)]~i. When the point a is within the region bounded by Cy and F (z) is any function which is regular and analytic in the closed inner realm of (7, we have the inequality where 8 is the least distance of the point a from the curve C. To prove this we remark that Cauchy's theorem gives and so | F (a) \ < ~ f ~-(z) ds < ~ [ \F (z) I da. ' ' 2n Jc z — a 277-8 Jp ' ' In the special case when F (z) = \Gn (z)]2 the inequality gives This is true for all polynomials Gn (z), and so, in particular, 27r8 Since 8 is independent of n, this inequality establishes the convergence of the series K(a,a)= |P0(a)|2 + | Pl (a) |2 + ... for the case in which the point a lies in the region bounded by C. Again, we have the inequality \Kn (a, 2) |2< [|P0 (a) || Po (z) | + | Pl (a) \\ P,(z) \ + ...]»< Kn (a, a) Kn (z,z), and if Rnm (a, z) denotes the remainder Rnm (a, z) = P^TF) Pn+l (z) 4- ... Pn+m (a) Pn+m (z), we have | Rnm (a, z) |2 < Rnm (a, a) Rnm (z, z). Since the series K (a, a) is convergent when a lies within C we can find a number N (a) such that if n > N (a) we have for all values of m | Rnm (a, a) | < e, where e is a small positive quantity given in advance, hence if N is the greater of the two quantities N (a), N (z), we have for n > N This establishes the convergence of the series K (a, z) - KM Pft (z) + Pn(a) P, (z) + .... Series of Polynomials 319 To prove that the series is uniformly convergent in any closed realm R lying entirely within C we note that the quantities Kn (a, z) are uniformly bounded in the sense that | Kn (a, z) |2 < Kn (a, a) Kn (z, z) < The general selection theorem of § 4-45 now tells us that from the sequence Kn (a, z) we may select a partial sequence of functions which converges uniformly in R towards a limit function/ (z). Since, however, the sequence converges to K (a, z) this limit function/ (z) must be identical with K (a, z) and so the series which represents K (a, z) converges uniformly in R, a and z being points within R. We now consider the integral = [ J Kn(a,z)-X{7'(z)}l\*ds, c where A is a constant which is at our disposal. We have Kn(a,z) \2ds = Kn(a,a), \ \y'(z)\ds= |2"d0=2,r, JC JO Kn (a, z) {/ (z)}* ds = Kn {a, g o I V f ^f — • I J^ < d where £ = e". Jo Now the function Kn {a, g (£)} vV (0 is regular and analytic for | £ | < 1, and so the last integral is equal to n {a, g (0)} V) = ^Kn (a, a) [/ (a)]-. Choosing A - ~- [y (a)]*, we have finally Jn = ^- | / (a) | — ^Tn (a, a). ZTT Our object now is to show that lim Jn - 0. n->oo Let us write £(£) = fo' (£)]*• Since gr' (£) ^ 0 for | £ | < 1, a branch of L (£) is a regular analytic function for | f | < 1. We now consider the set of analytic functions E (£) regular in | £ | < 1 and such that f2ir |L(£)tf(0|»<M=l. Jo 320 Conformal Representation Let a be a fixed number whose modulus is less than unity, then the maximum value of | E (a) \ 2 is [1- |«|2]-i|L(«)|-2. To see this we put L(QE(l) = t0 + tlt+...+tnF+...9 then on the above supposition and Schwarz's inequality gives \E(a)\*\L(a)\*< £ \tn The sign of equality holds when, and only when, tn = ca", n=0,l, 2, .... that is, when L (£) # (£) = j-^^g. M f2' ^ _ 271L w Jo 1 1 ~s£i»~ r-i«i2> therefore 27r | c |2 = 1 - | a |2, and so JB(0-.(1-* -.-— -. 1 - Jo (27T)*Zf(C) Now let £ (0 = -E {y (2)} = (? (z) = (? {g (J)}, then jE? (^) and (7 (2) are simultaneously regular, and 1= fa'|L(0^(0|lde= [8"|i7'(OII^U)|'de= f IGW*. Jo Jo Jc Finally, E (0) - (7 (a), so that max | E (0) |2 - max | G (a) |2, therefore K (a, a) = max | G (a) |- = max | JB (0) |« = ^-/^ = ^^ | , and so Km -/„ = ^ | / («) | - ^ (a, a) = ~ {| / (a) | - | g' (0) |} = 0. n -> oo ^^ ^7r Since [^n (a, z) - A {y' (z)}Jp = Fn (z) is a regular analytic function in the closed inner realm of (7, we have for any point z0 within C whose least distance from C is 8, and so as n ->oo, lim | .Pn (z0) | = 0. Region Outside, a Closed Curve 321 Hence K (a, z,) = lim Kn (a, z0) = A {/ n->oo Furthermore, since K (a, a) - j^ | / (« we have / W - 2. I* o rg and so y (2) = ~ -r [K (a, z0)]2 dzQ. A (a, a) Ja If the curve (7 instead of being of unit length is of length I the ortho- gonal polynomials Pn (z) are defined so that and the general formula for the mapping function becomes where e is a number with unit modulus and is equal to unity when the mapping function is required to be such that y' (a) > 0. A study of the expansion of functions in series of the orthogonal poly- nomials Pn (z) has been made recently by Szego and by V. Smirnoff, Comptes Rendus, t. CLXXXVI, p. 21 (1928). § 4-82. The mapping of the region outside C' . If we write z' (z — a) = 1, ww' = 1, the interior of C maps into the region outside a closed curve C' in such a way that the point z = a maps into the point at infinity in the z'-plane. The interior of the unit circle | w \ < 1 is likewise mapped into the region | w' | > 1, the point w = 0 corresponding to w' = oo. Hence the region | wr \ > 1 is mapped on the region outside C' in such a way that w' = oo corresponds to z' = oo, the relation between the variables being of type z' = rw' + T0 -f T! (w')~l + ... + rn (w/)~n -f .... Since w = y (z), the function which gives the conf ormal representation is w' = [y {a 4- (z')-1}]-1 = 0 (A say. Szego has shown that the function 0 (z') may also be obtained directly with the aid of an orthogonal system of polynomials Yln (z') associated with the curve C". If T = | T | 6fa, we have in fact the formula 322 Conformal Representation EXAMPLES 1. If z0 is a root of the equation Pn (z) = 0, prove that —^--- I zas. Z -Z0| Hence show that z0 lies within the smallest convex closed realm R which contains the curve C. 2. If C is a circle of unit radius, P (z} — zn L n \*f "~ * > 3. If the curve C is a double line joining the points — 1,1, the polynomial P (z) becomes proportional to the Legendre polynomial. Note that in this case the series K (a, z) fails to converge, but this does not contradict the general convergence theorem because now the points a and z do not lie within C. 4. If jRn (z) is any polynomial and a any point within the curve (7, prove that § 4-91. Approximation to the mapping function by means of polynomials Let a circle of radius R be drawn round the origin in the z-plane and let w = / (z) = ao + aiz + a2z* + ••• be a power series converging uniformly in its whole interior. This maps the circle on a region of the complex w-plane. For the area of this region we easily find the expressions A = I" [2?r I /' (z) \*rdrde (z - re«) Jo Jo = 277 \R Jo rdr S | an n=l al I2 + TT S n\ an |2 R2n + ... . n-2 The area of the image region is always greater than 77.R2 | ax |2 when ax 7^= 0 and is always greater than TTH | an |2J?2n when an ^ 0. When the mapping function/ (z) is such that a: = 1 the result is that the area of the picture is greater than that of the original region unless the picture happens to be a circle of radius R. Suppose now that we are given a simply connected smooth limited region B of the z-plane. Let dr be an element of area of this region, we then look for a function / (z) regular in B which makes the integral 7 =JJ |/'(S) |«(*T * L. Bieberbach, Rend. Palermo, vol. xxxvm, p. 98 (1914). Approximation by Means of Polynomials 323 as small as possible. To make the problem definite we add the restrictions that / (0) = 0, /' (0) = 1, and that / (z) is a polynomial of the nth degree. These conditions are satisfied by writing f(z) = z + a2z*+ ...+ anzn. ...... (A) If / (z) -f €g (z) is a comparison function we have to formulate the conditions that the integral (0 = j] I /' + *g' (z) \2dr^ [/' (z) + eg' (z)] [T(z) + eY~&} dr may be a minimum for € = 0. These conditions are ~ = g' {z) *rw dr = I v' M I2 dr- The inequality is always satisfied, but the two equations are satisfied for all forms of the polynomial g (z) only when the coefficients as satisfy certain linear equations. If where z is the conjugate of z, these equations are 2z0>1 4- 4zltla2 + 6z2jla3 4- ... 2nzn_lt lan = 0, i + (n.%) Zi,n~ia* + (n.'3) z^^a^ + ... (n.n) zn_^n^an - 0, and 2zlf04- (2.2)z1>1a2-f (2.3)z1,2a3+ ... (2.n)zltW.1oB = 0, ^n-ifo + (ra.2) zn_1>]La2 -f (w.3) 2;n_1>2a3 + ... (n.n) zn_lin^dn = 0. These linear equations are associated with the Hermitian form n2ln2 (p+ 1) (5+ l)^a^, p=0 g«0 and possess a single set of solutions for which / is a minimum. By giving different values to n we obtain a sequence of polynomials which in many cases converges towards a limit function F (z). The question to be settled is whether this function F (z), among all mapping func- ^ -\ tions with the properties F (0) = 0, F' (0) = 1, gives the f \ smallest possible area to the picture into which B is I ^ \ mapped. The following simple example tells us that this is not always the case. Consider the region B which arises from a circle when the outer half of xme of its radii is added to the boundary (Fig. 26). There is no Fi&- 26' polynomial which maps this region B on a region of smaller area. For 324 Conformal Representation by means of a polynomial the region B is mapped on another region which has the same area as the region on which the complete circle is mapped and, unless the polynomial is simply z, this region has an area which is greater than that of the circle. Hence in this case all minimal polynomials are equal to z and F (z) is also equal to z. Bieberbach has investigated the convergence of the sequence of polynomials to the desired mapping function for the type of region discovered by Carath^odory*. For such a region the boundary is contained in the boundary of another region which has rib point in common with the first. The interior of a polygon is a particular region of this type and so also is the interior of a Jordan curve. Bieberbach's method of approximation has been used recently in aerofoil theory for the mapping of a circle on a region which is nearly circular f. Introducing polar co-ordinates, z = reie, and supposing that on the boundary r = 1 -f y, where y is small, we may write T2TT fl+y I r2ir m }Q Jo 2> 4- <7 + 2 J o Hence retaining only terms up to the second order in the binomial expansion of (1 + y)p+<J+2? ZPQ= \ y-cos (p - q) d.dO + P -f y2.cos (p - q) 0.d0 Jo * J o (p - q) d.dd + -- 2.sin (p - q) 0.d0, p These quantities may be determined from the profile of the nearly circular curve when this is given. Now writing zw - fw + irj^, ap = <f>p + i$p, where fOT, 77^, <f>p and ifjp are all real, and neglecting all the coefficients after a4 in the expansion (A), we obtain the following equations for the determination of </>2 , </>3 , </>4 , i/j2 , 03 , i/r4 : £01 + 2£ii</»2 + 3£12(/>3 -f 3^^ -h 4|13</>4 -f 47?13</r4 = 0, ^w + 2fu^a - 3^12^3 -f 3|12</r3 - 47?13^4 + 4^1304 = 0, ^02 + 2^12^2 ~ 2^^ + 3f22S63 -f 4£23<£4 + 4^23^ - 0, ??02 + 2^12^2 + 2^12^2 + 3f22</T3 ~ 47723(^4 + 4^23^ - 0, %j + 27713^2 + 2£13</r2 -f 37723^63 + 3^^ -f 4fa * Jtfo^. ^Inn. vol. LXXII, p. 107 (1912). t F. Hohndorf, Zeite. f. any. Math. u. Mech. Bd. vi, S. 265 (1926). The conf ormal representation of a region which is nearly circular is discussed in a very general way by L. Bieberbach, Sitzungsber. der preussischen Akademie der Wissenschajten, S. 181 (1924). DanielVs Orthogonal Potentials 325 Eliminating 04 and </r4 we obtain the equations (0133) + 2 (1133) fa +3 (1233) fa 4 3 [1233] </»3 = 0, [0133] + 2 (1133) fa - 3 [1233] <£3 + 3 (1233) ^3 - 0, (0233) + 2 (1233) fa - 2 [1233] 02 + 3 (2233) <£3 - 0, [0233] + 2 [1233] </>2 4 2 (1233) 02 4-3 (2233) </r3 - 0, where (jpgr*) - $„$„ - ^ fw - rjprTjqs, [pqrs] = 17^17,., 4 f^ifc, - i?prfw, and these finally give the values vNfa = Z2,_3 , yAty, - Z2,_, , v = 2, 3, 4, where N = (1233)2 4- [1233]2 - (1133) (2233), Zl = (0133) (2233) - (0233) (1233) - [0233] [1233], Z2 - [0133] (2233) 4 (0233) [1233] - [0233] (1233), ZB - [0133] [1233] 4 (0233) (1133) - (0133) (1233), Z4 - [0233] (1133) - (0133) [1233] - [0133] (1233). § 4- 92. DanielVs orttiogonal potentials. Consider a set of polynomials pQ (z), Pt (z), ... defined by the equations* (0,0) (1,0) (71,0) (0, 1) (1, 1) (n, 1) where (0, 71-1) (1,73 1 (0,0) (0,1) (0,71) (1,0) (1,1) (l,w) (71,71- zn and (n,0) (TI,!) (7i, n) (m, n) = ^- zmzndr, the integral being taken over the region to be mapped on a unit circle. A denotes here the area of the region and dr an element of area enclosing the point z. These polynomials satisfy the orthogonal conditions \ r f _ 3 Jj Prn (z) Pn (*) dr - 0, m^n = 1, m = n. A mapping function / (z) which satisfies the conditions / (a) = 0, /' (a) = 1, is given formally! by the expansion 4 W f J a Pl (z') where fif = p0(a) p0 (a) + ± (a) 4 .... * These equations are analogous to those used by Szego. f This series does not always represent an appropriate mapping function as may be seen from a consideration of the circular region with a cut extending half-way along a radius as in § 4-91. 326 Conformal Representation To see this we write /' (z) = flo^o (z) + aiPi (z) 4- a2p2 (z) 4- ... , (z) where a0 , ax , . . . ; a0 , ax , ... are coefficients to be determined, so that the integral f (z)f'(z).dr = a0a0 + a,at -f ... may be a minimum subject to the conditions /'(a)=l, />)=!. ...... (A) Differentiating with respect to a0, al9 ... ; a0, al9 ... in turn we find that «n = &Pn («), W = 0, 1, where i and ^ are Lagrangian multipliers to be determined by means of the equations (A). We easily find that 1 = kS, kS = 1, and so Sf (z) = Po(a) Po (z) + ^1S) Pl (2;) + ... . If pn (z) = un ~ ivn , where un and vn are real potentials which can be derived from a potential function <f>n by means of the equations _ d<f,n ty, M" ~ dx ' " ~ dy ' we have U[ (-^ ^ + ^ ^ dr = 0, m^« J[ J J \ 3x 3o; dy dy / = 1, m = 7i. The potentials </>0, </>1? ... thus form an orthogonal system of the type considered by P. J. Daniell*. This definition of orthogonal potentials is easily extended by using a type of integral suggested by the appropriate problem in the Calculus of Variations. For the unit circle itself the orthogonal polynomials are Pn(*) = &.(n+ 1)*, and the mapping function is consequently given by the equations f(z\ = v1 - au) vz - <*) § 4-93. Fejer's theorem. Let be the function mapping a region D in the Z-plane on the unit circle d with * Phil. Mag. (7), vol. ii, p. 247 (1926). Fejer's Theorem 327 equation | z \ < 1 in the z-plane. We shall suppose that D is bounded by a Jordan curve C and that Z = H (9) is the point on G which corresponds to the point z = eie on the unit circle c which bounds the region d. At this point, if the series converges Z = aQ + a^19 -f a2e2ie + ... anein* + ... = ^o + Wi + ^2 + ••• say. ...... (2) Now by Cauchy's form of Taylor's theorem = 0, n < 0, where n is an integer and the contour is a simple one enclosing the origin and lying within the circle of convergence of the power series. On account of the continuity of / (z) in d we may deform the contour until it becomes the same as c without altering the values of the integrals. Hence, writing £ = em we get f2jr foo 2?7 an = H (a) e~ina n > 0, 0 = // (a) einada. Jo Jo These equations show that the series (2) is the Fourier series of the continuous function H (6) and is consequently summable (<7, 1) (§ 1*16). Now consider a circle j z \ = p where p< 1. The function / (z) maps the interior of this circle on the interior of a region R whose area A is, by § 4-91, equal to the convergent series 7r[|a1|V+2|a2|V+-"]- This area A is bounded for all values of p and is less than B, say, •'• "[KIV + 2 I «2lV + ...n\an\*p*n]<B for 0 < p < 1 and so 7r[|a1|a+2|o2|2 + ...n\an\*]<B. This inequality shows that the series S n \ an |2 is convergent. Now this property combined with the fact thfrt (2) is summable ((7, 1) is sufficient to show that the series (2) converges. Writing sn - UQ + u^ + ... un, (n + 1) Sn = 50 + $! + ... sn we have Sn = sn - an where (n + 1) crn = t^ + 2^ + ... nun. It is suffi- cient then to show that an -> 0 as /i -> oo. With the notation #n = | i^n | we have the inequality [(m f- 1) vm+1 -f ... (m + p) vm + P]2 < [(m + 1) + ... (m + p)] [(m + 1) v2m+1 + ... (m + p) v*m+J>]. The first factor on the right is less than 1 -f 2 4- ... (m + p) which is less than (m + p + I)2. Also, since the series 2rwn2 converges we can choose 328 Conformal Representation m so large that the second factor is less than e2 whatever p may be. We may now write n = m + p, * = v\ + 202 + ••• mvm (w + 1) vm+l + ... (m + p) vm+J> °n ~~ m -}_ p -j- i m _+_ p -f 1 > where the second term on the right is less than e and since p is at our disposal we may choose it so large that the first term on the right is less than e. Hence we can choose n so large that | vn \ < an* < 2e and so | an | -> 0 as n -> oo. It follows then that the series (2) converges and that the co-ordinates (X, Y) of a point on a simple closed Jordan curve can be expressed as Fourier series with 6 as parameter. The theorem implies that the series (1) converges uniformly throughout d and that the mapping by means of the function / (z) may be extended to regions which are slightly larger than d and D. Reference is made to Fejer's papers, Milnchener Sitzungsber. (1910), Comptes Rendus, t. CLVI, p. 46 (1913) for further developments. Also to the book by P. Montel and J. Barbotte, Lemons sur les families normales de fonctions analytiques (Gauthier-Villars, Paris, 1927), p. 118. CHAPTER V EQUATIONS IN THREE VARIABLES § 5-11. Simple solutions and their generalisation. Commencing as before with some applications of the simple solutions we consider the equation _ p W ~ ^ \d& + of the propagation of Love-waves in the direction of the #-axis. If now p and fjb have the values p0 , /ZQ respectively for z > 0, and the values p± , p^ respectively for z < 0, it IP useful to consider a solution of type v = v0 = (A cos sz -f B sin sz) sin K (x — qt)3- z > 0, v = Vl = Cfcto sin K (x — gtf), 2 < 0, where the constants are connected by the relations A»*V = Mo (*2 + *2)> Pi^V = Mi (^2 ~ *a)- In the expression for v2 we take h > 0 so that there is no deep pene- tration of the waves. The boundary conditions are /^ -~ = 0, when z = a, These equations give 4 sin sa = J5 cos sa, 4 = (7, Putting ^ = C02p0 , ^ = q^! we have with s = K cosech o>0 , A = K sech o^ , c0 = g tanh o;0 , cx = y coth cox , cosh Wi = — sinh o>0 cot (CLK cosech co0) = ( 1 — ^ tanh2 o>0 ) , Mo V ci ' and it is readily seen that there are no waves of the present type unless c0<c1. Matuzawa has examined the case of three media arranged so that in his notation v = vl = Al (e°iz + e-*iz) cos (pt + fx), p = p1? M = Mi> 0 > 2 > - A, v = vz = (^2e**z + Be~'*z) cos (jp£ -f /»), p = />2J M==M2> — h> z> — H, V = V3 = ^36*8* COS (p^ -h /#), P = P3> M^MSJ "" H > Z. 330 Equations in Three Variables The boundary conditions dvl dvz . , ^i - v2, to - = ft ~ at z - - give ^4j (e~*iA + Eliminating ^ij, ^42» ^3> ^2 and writing rx = tanh sji, r2 = tanh s2h, T2 = tanh s2H, we obtain the equation ? (T2 - T,) The cases ^, 52, «§3 all real and 52 imaginary, sl9 s3 real, are not com- patible with an equation of this type. When s2 is real it appears that there is only one value of sl and this is an imaginary quantity; when .$2 is an imaginary quantity it appears that there are two possible values of sl and these are both imaginary. Matuzawa has examined the six possible cases A B G D E F Cl<C2< C3 Cl<C3< C2 C2<Cl< C3 C3 < G! < C2 C2<C3< Ct C3 < C2 < Cx and concludes that in cases B, D and F there is no solution. § 5*12. The simple solutions considered so far correspond to the case of travelling waves. We shall next consider a case of standing waves and shall take the equation of a vibrating membrane dt2 ~~ \dx* Let the boundary of the membrane consist of the axes of co-ordinates and the lines x =• a, y = b. The expression . m-rrX . UTTV , 4 ^ . .. w = sin — — sin -^ {^4mn cos pt + Bmn sin jrf} w c/ satisfies the condition w = 0 on the boundary and is a simple solution of the differential equation if This equation gives the possible frequencies of vibration, ra and n being integers. A more general type of vibration may be obtained by summing with respect to m and n from m = 1 to oo and n = 1 to oo. Standing Waves 331 The resulting double Fourier series is usually a solution which is sufficiently general to make it possible to satisfy assigned initial conditions w ^ WQ' dt = ^° f°r ^==0> by using coefficients Amn, Bmn determined by Fourier's rule 4 4 [a f& . mrrx . a 6 nny - 4 fa f6 . mTTX . n = ~T— wo sin s a6jpJoJo a EXAMPLE Find the nodal lines of the solutions . . TTX . *rrtj f nX 7rtJ\ > = A sm - - sm -- (cos h cos — ) cos pt, a a \ a a / ^ A . '27TX . 2rry w = A sin — sin — - cos pt, a a ~ ( . SKX .Try . TTX . STT^I w — C {sin — sin - — sin — sin — -> coapt, \ a a a a j • which are suitable for the representation of the vibration of a square if p has an appropriate value in each case. § 5-13. Reflection and refraction of electromagnetic waves. In a non- conducting medium the equations of the electromagnetic field are* curl H - J9/c, div D = 0, curl E = - B/c, div B = 0, and the constitutive relations are D - xE, B= /*//, where the coefficients K and ^ can be regarded as constants if the material is homogeneous and the frequency of the waves is not too high. If all the field vectors are independent of z, their components satisfy the two-dimensional wave-equation where V2 = c2/Kp, = 1/s2, say. The permeability of all substances is practically unity for frequencies as great as that of light. Hence for light waves it is permissible to write V = C/VK, and in this case we may also write /c = n2, where n is the index of refraction of the medium. Let us now suppose that the medium with the constants (K^, /^) is on * For convenience we denote a partial differentiation with respect to the time t by a dot. 332 Equations in Three Variables the side x < 0 of the plane x = 0, and that on the other side of this plane there is a medium with constants (/c2, /x2). We shall suppose that when x < 0 there is an incident and a reflected wave, but that for x > 0 there is only a transmitted wave. We shall suppose further that the electric vector in all the waves is parallel to the axis of 2, then with a view of being able to satisfy the boundary conditions we assume Ez - A& + AM (x < 0), E, = A2e2 (x > 0), where e^, e± ', e2 denote respectively the exponentials g __ gitofs^j; co8#i + 1/ sin <£>!)-'] g ' The corresponding expressions for the components of H are Hx - (csjfr) (A& 4- AM) sin </>!> # < 0, Hx = (c52//LL2) ^4262 sin </>2> a; > 0, //„ = (c$i/ /L4) (A'e/ ~ ^iei) cos </>!, a < 0, Hy = — (cs2/iJL2) A2e2 cos </>2, a; > 0. The boundary conditions are that the tangential components of E and // are to be continuous. These conditions give A.} -f- AI = A2, sin fa.ptSi (Al -f AI) = /^i«§2^2 sin ^2» COS <>i.X2<Sl (-^1 — ^/) = ^152^2 COS <2- TT l l2 2 l Hence . ^ = ^- = — when ux = u2. sin (f>2 p,2si ni This is the familiar relation of Snell. Writing A9 = fi-4, ^2 = T^, where 7? and T are the coefficients of reflection and transmission re- spectively, we have 1 — R _ sin ^ cos fa T* _ gin (^i — ^2) 1 -}- jR sin <^2 cos </>! ' sin (0t + ^2) ' 2 — T sin <^x cos ^>2 ^ 2 sin ^>2 cos ^ 27 ~" sin ^2 cos </>! ' ~~ sin (fa 4- <^2) In the case when the electric vector is in the plane of incidence we write EX - - (C& + CM) sin <£i> EX = - ^2^2 sin </>2, ^ = ((7^ - CW) cos ^ (a; < 0), Ev --= ^63 cos <£2 (x > 0), H, = (c^/^) (OlCl + CM), Hz and the boundary conditions give — GI) cos ^ = (72 cos ^2 -f Cf1/) sin </>! = ^^ sin Reflection and Refraction 333 The third equation implies that the ^-component of D is continuous at x = 0. These equations give sin <A1 = Sin Thus Snell's law holds as before. If we further write CS-pCv Ct=rCt, so that p, T are the coefficients of reflection and transmission respectively, WC haVe = tan (A - c£2) p tan (^ + </>2) ' 2 sin <£2 cos (/>! r ~ sin (<£j, + <f>2) cos (</>! - ^>2) ' Of the four quantities R, T, py r only one can vanish, viz. the polarizing angle Ox is defined as the angle of incidence for which p = 0. This angle is given by the equation tan (^ -f- <f>2) = oo, and so tan Oj — n2/n1 . When the incident light is unpolarized it consists of a mixture of waves in some of which E is parallel to the axis of z and in the others H is parallel to the axis of z. When such light strikes the surface x =0 at the polarizing angle the waves of the second kind are transmitted in toto, and so the reflected light consists merely of waves of the first kind and is thus linearly polarized. Reflection and refraction of plane waves of sound. Consider a homo- geneous medium whose natural density is p0. When waves of sound traverse the medium the density p and pressure p at an arbitrary point Q (%> y, z) have at time t new values which may be expressed in the forms p = Po(l + 5), p = p0(l 4- As), where p0 is the undisturbed pressure and A is a coefficient depending on the compressibility. The quantity s is called the condensation and will be assumed to be so small that its square may be neglected. We now suppose that the velocity components (u, v, w) of the medium at the point Q can be derived from a velocity potential <f> which depends on the time. Bernoulli's integral [dp d<f> , . \ - + -7T7 = constant ] P dt then gives the approximate equation where c2 = pQA/pQ = (dpjdp)Q is the local velocity of sound and is constant since A and pQ are constants. The equation of continuity 334 Equations in Three Variables and the equations u ~ ~ , v = -~ , w = ^, when ?/, t;, w are small, give the wave-equation <>i The conditions to be satisfied at the surface separating two media are that the pressure and the normal component of velocity must be con- tinuous. On account of Bernoulli's equation the continuity of pressure p / implies that p ^~ is continuous. Let us now consider the case in which two media are separated by the plane x — 0. We shall suppose that in the medium on the left there is an initial train of plane waves represented by the velocity potential ^ = aoC«»«-f»-«>, and that these waves are partly reflected and partly transmitted. We therefore assume that (/>! = a0etn(t-tx-™} + a1ein(e+ffl?-1iy), x < 0, </>2 = a2etn(t-Sx-™} , x > 0. The boundary conditions give Pi (0'0+ %) where pl and p2 are the values of the natural density for x < 0 and x > 0 respectively. If cx and c2 are the two associated velocities of sound, we must have c^cos^, c^sin^, c2 £ = cos «2 > C27? ^ sin a2 • Therefore __ cos «! — clpl gos «2 ^2 c°t ai ~" Pi 1 ° COS al + ClPl COS ^2 ° P2 CO^ al + Pl cos % _ 2/>! cot " - " ~ " — — ri . . COS «! -h C1pl COS a2 /02 COt «! -f- /)j COt a2 The equation c2 sin % = ct sin «2 gives a law of refraction analogous to Snell's law. When the second medium ends at x = 6, where b > 0, and for x > b the medium is the same as the first, there are three forms for the velocity potential: ^ ^ ^ginu-f*-™) + a^in(t^x-^y^ x < 0, </>2 = a2ein(t"^-^ + a3einU+^-^>, 6 > a; > 0, (/>3 - a4emU-^-^>, and the boundary conditions give pi K -f «i) == p2 K + «3)» ^ K - «i) = £ («2 - %)> Energy Equation 335 Therefore cos (*6£) + -^ sin 4 ; sin (n&£) 4- ^ 4 = « os (n£) + Jt i2 + sn £-2 - M sin bPl £P2J It should be noticed that these equations give Kl2- I oil2- Kl2, I 19 I 19 I 19 p2 1 «2 12 - />2 1 % r = PI I ^ r cot a , and the first of these equations indicates that the sum of the energies per wave-length of the reflected and transmitted waves in the first medium is equal to the energy per wave-length in the incident wave. It should be noticed that if sin (nb£) ^ 0, the condition for no reflected wave (% = 0) is £p2~ £pi> and is independent of the thickness of the second medium. We have assumed so far that there is a real angle «2 which satisfies the equation c± sin 0% — c2 sin «x , but if c2 > Cj it may happen that there is no such angle. If the value of sin «2 given by this equation is greater than unity, cos «2 will be imaginary and the solution appropriate for a single surface of separation (x — 0) will be of type </>x - a0ein(t^x-^ -f a1etnU+^-Tjy), x < 0, ^2 = o2et"n(*-1»')-*a!, x > 0, 0 > 0. In this case there is no proper wave in the second medium, and on account of the exponential factor e~6x the intensity of the disturbance falls off very rapidly as x increases. The corresponding solution of the problem for the case in which the second medium is of thickness b is obtained from the formulae already given by replacing £ by — id. It is thus found that P* -- LP2 0J a4 [cosh (nbO) 4- \i (^ - ^} sinh (nb0) I , L Wl Sp2/ J 2 -f ^l sinh (nbO). The coefficient a3 of the disturbance of type einu"liv)+a* which increases in intensity with a; is seen to be very small so that this disturbance is small even when x = 6. 336 Equations in Three Variables In the present case the reflection is not quite total, for some sound reaches the medium x > b. The change of phase on reflection is easily calculated by expressing aj/a0 in the form jRe*". Let us now consider briefly the case when nb£ — krr, where k is an integer. In this case sin (nb£) = 0; there b no reflected wave and the formulae become simply </»i = a^ein(t~^-^\ x < 0, </>2 •= «o Wp2) ein(t-*x-™> + a3ein(t~™> sin (£nx), b > x > 0, (f>3 = a0etnU~f*-'I1'), x > b. It will be noticed that the value of n is precisely one for which there is a potential </> fulfilling the conditions -~ = 0 for x = 0 and x = b. The slab of material between # = 0 and x = b can be regarded as in a state of free vibration of such an intensity that there is no interference with the travelling waves. The absorption of plane waves of sound by a slab of soft material has been treated by Rayleigh* by an ingenious approximate method in which the material is regarded as perforated by a large number of cylindrical holes with axes parallel to the axis of x and the velocity potential within these holes is supposed to satisfy an equation of type where h is a positive constant. The new term is supposed to take into consideration the effect of dissipation. At a very short distance from the mouth (x = 0) of a channel it is £}2JL PV2JL assumed that the terms ~ -2 and ^ ^ may be neglected and that the solution is effectively of type £ = eint {a! cos k'x -4- 6' sin k'x}, where c2k'2 = n2 — inh. If the channel is closed at x = 6, we have ^ - = 0 there, and so we may write </» = ^4Vn'cosfc' (x— b). When x is very small C25 = _ _r = _ inA'eini cos (k'b), u ik' c*s n {k'b}- * Phil. Mag. (6), vol. xxxix, p. 225 (1920); Papers, vol. vi, p. 662. Reflection 337 If, for x < 0, we adopt the same expression as before, viz. we have Now let a be the perforated area of the slab and v' the area free from holes. The transition from one state of motion on the side x < 0 to the other state on the side x > 0 is assumed to be of such a nature that (cr -f a') ul= cm, These equations give the relation a0 — a* ikf .-. ,, . , tan (kb) a0 ai n$ <* for the determination of the intensity of the reflected wave. When h — 0, we have | ax | = | a0 | and the reflection is total, as it should be. When a = 0, a± = a0, and there is again total reflection. On the other hand, if a = 0, the partitions between the channels being infinitely thin, we have, when h = 0, * _ ng cos (k'b) — ik' sin (k'b) __ cos «j cos (k'b) — i sin (k'b) 1 ~ ° ?i£ cos (&'6) -f ^' sin (k'b) ~ ° cos «j cos^F6) 4- * sin (k'b) ' In the case of normal incidence «x = 0, ax = a0e~2ifc/b, and the effect is the same as if the wall were transferred to x = b. When h is very small but the term k2 in the complex expression k' = k^ + ik2 is so large that the vibrations in the channels are sensibly extinguished before the stopped end is reached, we may write cos (ik2b) = Jefc2&, sin (ik2b) — \iek^, tan (k'b) = — i, and the formula becomes a0 + di (cr + a') cos a±' EXAMPLES 1. In the reflection of plane waves of sound at a plane interface between two media the velocity of the trace of a wave-front on the plane interface is the same in the two media. * [Rayleigh.] 2. When the velocity of sound at altitude z is c and the wind velocity has components (u, v, 0), the axis of z being vertical, the laws of refraction are expressed by the equations <f> = <t>Q , c cosec 9 -h u cos <f> + v sin <f> =-- c0 cosec 00 + UQ cos <£0 + v0 sin <£0 = A, say, where (0, <f>) are the spherical polar co-ordinates of the wave-normal relative to the vertical polar axis and the suffix 0 is used to indicate values of quantities at the level of the ground. 3. Prove that the ray- velocity (the rays being defined as the bicharacteristics as in § 1*93) is obtained by compounding the wind velocity with a velocitj7 c directed along the wave-normal. See also Ex. 1, § 12-1. 338 Equations in Three Variables 4. The range and time of passage of sound which travels up into the air and down again are given by the equations rz x = 2 (c2 cos <£ -f us) dz/T, J o fZ y = 21 (c2 sin <f> + vs) dz/r, J o fZ * = 2 sdz/r, Jo where s — A — u cos <f> ~ v sin <f>, r — s (s2 —c2)*, and Z is defined by the equation s = c. § 5-21. Some problems in the conduction of heat. Our first problem is to find a solution of the equation which will satisfy the conditions 6 — exp [ip (t — x/c)] when y = 0, 6=0 when y = oo. Assuming as a trial solution exp [ip (t - &/c - y/b) - ay], Kip\ 2 7?2~l a + ~-j — ^-g • Therefore 6 = 2a/c, p2 ( ~ -f ~) = a2. -r ^2 cay The result tells us that if the temperature at the ground (y = 0) varies in a manner corresponding to a travelling periodic disturbance, the variation of temperature at depth y will also correspond to a periodic disturbance travelling with the same velocity but this disturbance lags behind the other in phase and has a smaller amplitude. The solution may be generalised by writing b = c tan </>, a = (c/2/c) tan <£, p = (c2/2/c) tan (/> sin 0, 7T f2 # = f (<f>) d<f>. exp [(ic/2/c) (c£ — x) tan <f> sin <^ — (cy/2/c) (tan (/> + i sin ^)], Jo where c i^> regarded as a constant independent of (f> and / (c/>) is a suitable arbitrary function. If we wish this solution to satisfy the conditions 0 = g (ct — x) when y = 0, 0 = 0 when y = oo, the function / (<£) must be derived from the integral equation f = Jo 2*) tan <f> sin </>], (— oo < u < oo). Solution 'by Definite Integrals 339 When the function g (u) is of a suitable type, Fourier's inversion formula gives foo / (<£) ^ (c/47T/c) (1 4- sec2 </>) sin <f>.g (u) du.exp [— (icu/2f<) tan </> sin </>], J-oo 0 < </> < 7T/2. In particular, if </ (tf) = (2K/cu) sin [tan a sin a (c^/2/c)], where a is a constant, we have 6 = f "sin <f> (1 + sec2 0) dJ> . e~ <<*/2«>tan* Jo x sin [(c/2/c) {(cZ — x) tan </> sin </> — y sin </>}]. Another solution may be obtained by making c a function of <f> and then integrating ; for instance, if c = 2/c cos (/> we obtain the solution TT f2 0 = f ((f>) d<f>. exp [i sin2 <f> (2i<t cos (^ — x) — y (sin ^ -f i sin <f> cos <^)]. Jo It should be noticed that the definite integral 7T f2 0 (#, y,z,t)= f((f))d</). exp [i sin2 0 (2/c^ cos 6 — x) — z sin </> — iy sin <b cos <i] Jo is a solution of the two partial differential equations and is of such a nature that the function 9 (x, y, t) = 0 (x, y, y, *) is a solution of equation (A). It is easy to verify, in fact, that if 0 (x, y, z, t) is any solution of equations (B) the associated function 0 (x, y, I) is a solution of (A), for we have d*e a^^a2© a2© a2© a2© 3z2 + a*/2 ~ a*2 + dy* + dz2 dydz = 2 — = 1— =1 8^ dydz K dt K dt ' Again, if we take c = 2/c cot <f>, we obtain an integral IT (2 0 (x, y, t) = / (</>) d<f>.exp [i (2.Kt cos (f> — x sin </>) — y (I -f i cos <^>)], Jo which is a solution of (A), and the associated integral 7T (2 0 (x, y, z,t)=\f (<f*) d<f> . exp [i (2/c£ cos <j> — x sin <f> — y cos <£) ^- 2] Jo ...... (C) 340 Equations in Three Variables is likewise a solution of the equations (B). Indeed, if c is any suitable function of cf> the integral r = J (ct — #) tan </> sin </> — (c/2/c) (2 tan (f> + iy sin </>)] is a solution of the equations (B). It should be noticed that the particular solution (C) is of type 0 (x, y, z, t) = e~*F (x, y - 2Kt), where F (u, v) is a solution of the equation This indicates that if F is any solution of this equation, then the function 9 ^ y^ t) = \_vp {Xt y _ 2Kt) is a solution of the equation (A). This is easily verified by differentiation. Since there is also a solution 0 = erKtF (x,y)9 we have two different wrays of deriving a particular solution of the equation (A) from a particular solution of the equation (D). Since F (u, v) = J0 \/\u* -f v2] is a particular solution of equation (D) there is a certain surface distribution of temperature 6 ^ J0 V|z2 -f 4fc2*2], when y = 0, which is propagated downwards as a travelling disturbance gradually damped on the way, the velocity of propagation being 2/c. If, on the other hand, we take F (u, v) = cos mu.exp v [m2 — l]i, we obtain a distribution of temperature 0 (x, y, t) = e-» cos mx.exp {(y - 2*0 [m2 - 1]!}, m2 > 1 ...... (E) in which a periodic surface distribution is decaying at the same proportional rate at every point of the surface. If m2 < 2 the foregoing distribution gives 0 = 0 when y = oo. The periodic distribution now travels upwards with constant velocity c - 2* (m2 - !)*/[! - (m2 - 1)*], and the rate of damping at depth y is the same as that at the surface, but at any instant the temperature at this depth is a fraction exp[- 1 + (m2- 1)4] of that at the surface. When m2 = 2 there is a distribution of temperature 9 - e-2*< cos (x -v/2), which is independent of the depth but does not satisfy the condition* 0=0 when y = oo. When m2 > 2 the distribution (E) gives 6=0 when * In this case there is no solution of type 0 = e~2ltt Y (y) cos (x N;2) which gives the foregoing surface value of t and a value 0=0 when y = oo for Y" (y) — 0. Circular Source of Heat 341 y = — oo, and the material into which conduction takes place may be supposed to be on the side y < 0. In this case the velocity of propagation is c = 2* (m2 - l)4/[(ra2 - 1)* - 1], and the temperature at depth | y \ is at any instant a fraction exp - [(m2 - 1)1 - 1] of that at the surface. We have seen in § 2*432 that if 0 (x, y, t) is a solution of equation (A) then the function 9 -, is a second solution. If, in particular, we take the function 6 (x, y, t) = er« F (x, y), where F (u, v) satisfies (D), we obtain the solution If r2 = x2 H- i/2 there is a solution </> = ^e 4^ J0(arlt) ...... (G) depending only on r and t which at time t — 0 is zero at all points outside the circle r = 2a/c. When t > 0 the temperature at points of the circle is given by </> = t~lJ0 (2a2K/t). The circle can thus be regarded as a source of fluctuations in temperature which are transmitted by conduction to the external space. The total flow of heat from this circular source in the interval t = 0 to t = oo may be obtained by calculating the integral dt. Now = - J0 ( Also [^ dt (alt2} e/0' (2a2K/t) = - l/2*a, Jo r °° eft (a/J2) J0 (2a2*/£) - 1/2/ca. Jo Hence dt ( ^ } = — l//ca. Jo \^r)r^a< and so the total flow of heat from the circle is 342 Equations in Three Variables This is independent of a and so our formula holds also for a point source. The temperature function of a point source of " strength" Q is thus ^ = (ty*™*)-1^/4* ...... (H) while that of a circular source is )'ie J0 (ar/0- ...... (I) This result is easily extended to a space of n dimensions, thus in three- dimensional space the temperature function for a spherical source of strength Q is t)-*e * sm(ar/t)/(ar/t). ...... (J) The solution for an instantaneous source uniformly distributed over a circular cylinder has been obtained by Lord Rayleigh* by integrating the solution for an instantaneous line source. The result is r2 + a2 - 2ar cos 0 r2 -f a8 A more general solution is H + a2 Integration with respect to t from 0 to oo gives a corresponding solution of Laplace's equation and we have the identity dt I I \ o -*-"" I I ty ^ ft \ *• *» I /-k . I C* . I I * ^-- t€/, | (M) r > a. I The temperature 0 due to an instantaneous line doubletf of strength q may be derived by differentiating with respect to y the temperature </> due to an instantaneous line source of strength q. Since the latter is <f> - (g/4^Kt) e-'2/4"', we have 6 = (qy^TTKH^) e~r2lM. ...... (N) The temperature due to a continuous line doublet of constant strength Q is obtained by integrating with respect to t between 0 and t. Denoting this temperature by 0 we have 0 = 'flcft = (qylZtTKr*) ' ~ [e-*l*"] dt = 0 = f'flcft = (qylZtTKr*) f ' ~ [e-*l*"] Jo Jo at * Phil. Mag. vol. xxn, p. 381 (1911); Papers, vol. vi, p. 51. t See Carslaw's Fourier Series and Integrals, p. 345 (1906). The direction of the doublet is that of the axis of y. The doublet is supposed to be "located" at the origin. Instantaneous Doublet 343 This solution may be used to find a solution of (A) which takes the value F (x) when y = 0 and is zero when y = oo and when t = 0. If 9 is to be such that 9, ~- and -5- are continuous for y > 0, an appropriate expression for 9 is 00 rJ f (X-X* 1 yasecaa \ (0) & __ — _ \ / da F(x-\- i/tana)e 4l<t . - Tt] 2 The first integral evidently satisfies (A) if y > 0, and the second integral tends to F (x) as y -> 0 if F (x) is a continuous function of x. In the special case when F (x) = 1 the expression for 9 takes the form ^y* 8eca a ""~ a and can be expressed in the well-known form* 277-* e~v2dv, (P) where u2 = y2/4:t<t and w > 0. If the boundary y = 0 is maintained at the temperature F (x, t) the solution which is zero when y = oo and when Z = 0 is given by the formula / l'^' dt' o (^ - t Y There is a similar formula for a space of three dimensions. If 9 = F (x, y, t) when z = 0 and 0=0 when z = oo and when £ = 0, the appropriate solution is (B) In this case an element of the integrand corresponds to an instantaneous doublet whose direction is that of the axis of z. Let us next consider a case of steady heat conduction in a fluid moving vertically with constant velocity w. The fundamental equation is 'N/J / CJ2/J O2/3 (7(7 / O " \J " where K is the diffusivity. Writing 0 = ©e"*/2* the equation satisfied by 0 is V20 = A20, * The transformation from one integral to the other can be made by successive differen- tiation and integration with respect to w of the firrt integral. 344 Equations in Three Variables where A = W/ZK. A fundamental solution of this equation is given by 0 = where R2 = (x - £)2 -f (y - 7?)2 + (z - £)2, (& *?, £ constant). In particular, if £ = 77 = £ = 0, we have the solution 0 = Ar~le~^r, where r is the distance from the origin, and this corresponds to the solution 6= Ar-le*(*-r\ ...... (T) This solution has been used by H. A. Wilson* and H. Machef to account for the following phenomenon. If a bead of easily fusible glass (0) be placed a few millimetres above the tip of the inner cone (K) of the flame of a Bunsen burner, a sharply defined yellow space (SSf) of luminous sodium vapour is formed in the current of gas which is ascending vertically with considerable velocity. This space envelops the bead and broadens out in the higher part of the flame, as shown in Fig. 27. Provided the gas-pressure is not too high, the critical velocity of Osborne Reynolds, at which turbulence sets in, will not be exceeded even in these parts of the flame, so that the flow remains laminar, and the sodium vapour de- veloped from the bead is driven into the hot gas solely under the influence of diffusion. The fact that the vapour extends beneath the bead in the direction OA is proof of the high values of the coefficient of diffusion assumed at high temperatures, and at this point diffusion must be able to more than counteract the upward flow. Since an iso- thermal surface corresponds in the theory of diffusion to a surface of equal partial pressure, it is supposed that for suitable constant values of A and 8 the equation (T) represents the surface enclosing the sodium vapour developed from the glass bead. When K is small and w large, this surface approximates to the form of a paraboloid of revolution with the origin as focus. Mache obtains the solution by integrating the effect of an instantaneous source which is successively at the different positions of a point moving relative to the medium with velocity w. In fact Fig. 27. j-O )-i Jo where A = w/2/c. * Phil. Mag. (6), vol. xxiv, p. 118 (1912); Proc. Camb. Phil Soc. vol. xu, p. 406 (1904). | Phil. Mag. (6), vol. XLVII, p. 724 (1924). Diffusion of Smoke 345 A similar solution has been used by O. F. T. Roberts* to give the distribution of density in a smoke cloud when the smoke is produced continuously at one point, and at a constant rate. The case in which the smoke is produced continuously along a horizontal line at right angles to the direction of the wind is solved by integrating the solution for the previous case. § 5- 31. Two-dimensional motion of a viscous fluid. If (u, v) are the component velocities at the point (x, y) at time t, p the pressure at this point, the equations of motion, when the fluid is incompressible and of uniform density />, are du du du 1 dp + u + v = -- _£ ot ox oy p ox dv dv dv 1 dp _ + u+v== -- _*: + v V2?; , ot ox oy p oy while the equation of continuity is This last equation may be satisfied by writing ddf dJj <jj - ' ,j/ _ _ T_ U - <~\ ) ^ - ~f\~ j dx dy where if* is the stream-function, and if fccfc dx dy r is the vorticity at the point (x, y) at time t, we have or If s — xv — yu, we have ds dv du Ss dv du ~- = x x -- v ^- + VJ ^ = a; -^ -- y x~ — 9x dx y dx dy dy y dy d*s __ y*y _ d*u dv d*s _ 92y 9 9^2 - * 9^1 - y a^2 + 2 g^' 3^2 - x TT ^5 35 ds 1 / 3p 9^ Hence -^ + ^5-+ v 5- = -- (x -*r ~ 2/^" 3^ 3a; 3y p \ dy y ox If x2 + y2 = r2 we may write ds dv ds du ds ds f dv du\ u o~+ v 5" = M w a -- V-^~. 3x 3t/ V 3r 9r/ * Proc. jRoy. Soc. London, vol. orv, p. 640 (1923). 346 Equations in Three Variables If the flow is of such a nature that p depends only on r and v/u is independent of r, we have o- u since s = r vx , we have dr i . _r — r / __ ~ r Hence in the special case when ^ depends only on r, and the velocity is everywhere perpendicular to the radius from the origin, we have the d. ^rential equation This indicates that the velocity V = s/r satisfies the equation j*-Tj-fo- which is of the same form as the equation of the conduction of heat when the temperature 0 is of the form 0 = V cos 6. In the present case if/ and £ are related since they both depend on r and so the equation for £ is The equation satisfied by ^ is where / (t) is an arbitrary function of t. In the particular case when we have s - - (r2/2v^2) e-f2/4*«, F = - (r/ The total angular momentum is in this case TOO srdr = — o and is constant. The kinetic energy is on the other hand TTO (^ V*rdr = Trp/2^2. Jo This type of vortex motion has been discussed by G. I. Taylor* in connection with the decay of eddies. The corresponding type of vortex motion in which r _. j-i e-r«/4v< ias been discussed by Oseenf, TerazawaJ and Levy§. * Technical Report, Advisory Committee for Aeronautics, vol. I, 1918-19, p. 73. t C. W. Oseen, Arkiv f. Mat., Astr. o. Fys. Bd. vn (1911). J K. Terazawa, Report Aer, Res. Inst., Tokyo Imp. Univ. (1922). § H. Levy, PhiL Mag. (7), vol. n, p. 844 (1926). Decaying and Growing Disturbances 347 § 5-32. Solutions of the form iff = X (x, t) + Y (y, t). The condition to be satisfied is _ dy*dt dx lty» dy dx* ~ Differentiating successively with respect to x and y we get d2X^Y_d*Yd*X^ dx* dy* dy* dx* We can satisfy this equation either by writing X = xa' (t) + b (t), Y = yA' (t) + B (t), ...... (B) , ... d'X r U^&X d*Y r mi,92y /p. or by wnting -^ = [/t (f)]» ^ , ^ = [M (*)]• ^ ....... (C) The supposition (D) ...... ^ 11^ A A leads to -^--- = 0, -~T = 0. 3x4 3^/4 These equations follow from (C) if we put /u, (t) = 0. Solving equations (C) for X and 7 we get X = a(t) e**M + b (t) er*»™ + xc (t) + d(t), Y - A (t) e^«> 4- B (t) e-y*w + yC (t) + D (t). Substituting in the original equation and assuming that a (t), b (t), A (t), B (t) are not zero, we find that /x (t) must be a constant ^ and that the functions a, 6, c, A, B, G must satisfy the equations f - Cap,* = vap,*, ^b' -f primes denoting differentiations with respect to t. If the functions c (t) and C (t) are chosen arbitrarily, a(f), b (t), A (t) and B (t) may be determined by means of these equations when their initial values are given. In particular, if a = A = 0, c = C = 0, we can have 6 - Pe1**, B = u = p,yeVf*'*r~ILV9 v = — This represents a growing disturbance in which each velocity com- ponent is propagated like a plane wave. The pressure is given by the equation r/J/n F+J4_0 + 1 348 Equations in Three Variables The fluid may be supposed to occupy the region x > 0, y > 0. If so, fluid enters this region across the plane x = 0 (Q > 0) and leaves it at the plane y = 0 (P > 0). The amount entering the region is equal to the amount leaving the region if P = Q, the density p being assumed constant. If V = 0 and pn is the pressure at infinity (a: = oo, y = oo), we have P - ^oo = pfji*P*e2»2vt-*(x+v). The pressure is generally greater than pm and is propagated like a plane wave with velocity c == iiv A/2 = v — - — . ^ u — v Thus the velocity of a plane pressure wave in an incompressible fluid is equal to v times the ratio of the vorticity and the transverse component of velocity. When the motion is steady the equation to be satisfied is aer* (VIL -f C) + be-** (vji - C) + Aer* (vp> - c) + Be~™ (vp, + c) = 0, and we have four typical solutions : 0 = jps2 + ex + qy* + Cy + D, ifj = v^y + be-** + ex + d, iff = VIJL (x + y) -f Ae,™ -f- be~*x + d, i/j = Aer* -f vpx -f Cy + D. ^2 \r Returning to the first case we note that when -~ 2 = 0 the equations (B) do not give all possible solutions, for if X - xa' (t) -f- 6 (0, the original equation becomes Writing C7 = -=— 2- we have the simpler equation which possesses a solution of type £ s U - f V"*2' cos A [y - a (t)] o> (A) d\, Jo where o> (A) is a suitable arbitrary function. For the corresponding motion roo fJ\ 4> = aw' («) -f yc (*) + 6 (<) + c-*" cos \\y-a (t)} <a (A) -~, Jo A* r°° ^/A « - c (t) - c-'« sin A [y - a (<)] w (A) ^, Jo A v - Laminar Motion 349 This solution may be used to study laminar motion. The corresponding solution for the case in which the motion is steady is *? i/i - Kx + Pe v + #7/2 + Ry + flf, where P,Q, It, S, K are arbitrary constants. If J£ -> 0 while the coefficients P, Q, /?, $ become infinite in a suitable manner, a limiting form of the solution gives the well-known solution 0 - Ay* + Qy* + Ry + S. It may be mentioned here that an attempt to find a stream-function iff depending on a parameter s but not on t, and such that 1 ,14- *U *• // X led to the equation ^ - ^ = / (a;, j,) The conditions for the compatibility of this equation and ' seem to require / (#, y) to be a constant. By a suitable choice of axes the former equation may then be reduced to the form _ n dxdy~ ' and so 0 = JT (x, s) 4- 7 ( EXAMPLES 1. In the case when there is a radial velocity U and a transverse velocity F, both of which depend only on r and t and when the pressure p depends only on r and t, the equations for U and F are dV TrdV UV (d2F 1 dV 1 T71 3 a*' + ^ a~ ^ --- = v 1^~2 + ~ 3 ~ ~2 FT a" 5^ dr r ( dr* r dr r2 ) dr Hence show that F satisfies the equation F ~ where ^L is a constant. If a — X/2v, prove that there is a solution of type y ^r2<T+it-«-2e-r2/*vt9 and verify that the total angular momentum about the origin remains constant. 2. Prove that the equation for F is satisfied by a series of type y _ rn*-m 1 1 __ _ . , _ ™. ____ (r2/vt} V rt \ (1 + » - 2a) (n + 3) (T /vl> _ v ' ; ' 350 Equations in Three Variables and verify that when n = 1, m = 2, rf-'e- <«•"«»» (l + --"- (r*/M) + -^ 1 (rV4rf)* + ...1. ^ CT — I (7 — & 41 J This is a particular case of Kummer's identity F (a; y; a;)e"a:= ^(y — a; y; ~^), where F (a; y; ar) is the confluent hypergeometric function (Ch. ix). 3. Prove that there is a type of two-dimensional flow in which { = *V, and 0 is consequently of the form t = €-**** F (x, y), where F (x, y) satisfies the differential equation A = 0. Prove that in the latter case if a2 > 62 there is a growing disturbance which is propagated with velocity vk2/a, and show that Discuss the cases in which F = cos ax cos a u' CHAPTER VI POLAR CO-ORDINATES § 6-11. The elementary solutions. If we make the transformation x = r sin 0 cos <£, y = r sin 0 sin <£, 2 = r cos 0, the wave-equation becomes &W 2dW 1 3 / . 3]f\ _! __ _ "3r« r ~3r " r2 sin 0 30 \sm 30 ) + r2 sin2 0 9e/>2 c2 9*2 ~ This is satisfied by a product of type W=R(r)®(0)Q(<f>)T(t), ...... (I) Mm if + WcT = 0, sin i ar* r ar where k, m and n are constants. The first equation is satisfied by T = a cos (to) 4- 6 sin where a and 6 are arbitrary constants ; the second equation is satisfied by O = A cos m<f) 4- J5 sin m<£, where .4 and B are arbitrary constants. The third equation is reduced by the substitution cos 0 = p, to the form Its solution can be expressed in terms of the associated Legendre functions Pnm (/x) and Qnm (p) which will be defined presently. When k = 0 the fourth equation has the two independent solutions rn and r~(n+1), except in the special case when n = — (n 4- 1), i.e. when n = — £. Making the substitution w = r% R in this case we obtain the equation d*w 1 dw _ dr^^r dr ~ U' which is satisfied by w == C + I> log r, where (7 and Z> are arbitrary constants. 352 Polar Co-ordinates The fact that rn and r"n~l are solutions of the equation for R furnishes us with an illustration of Kelvin's theorem that if / (x, y, z) is a solution of Laplace's equation, then 1 f(x y, ?\ r J\r*> r2' r*) is also a solution. The transformation in fact transforms rn 0<1> into r~n~l 0O ; it also transforms r~i (G -f D log r) Q<J> into r~* (G — D log r) 0<I>. When m = 0 and n = 0 the differential equation for 0 is satisfied by Thus, in addition to the potential functions 1 and -, we have the potential functions 1 ! r + z ill r + z 2lo8r_i and 2rlogF-V It should be noticed that 3 /I. r + z\ 1 1 1 In fact we have ^- log (r + z) = - , and it is easily verified that log (r -f z) and log (r — z) are solutions of Laplace's equation. These formulae are all illustrations of the theorem that if W is a solution of Laplace's equation (or of the wave-equation), then is also a solution of Laplace's equation (or of the wave-equation). § 6-12. In the case of the wave-equation the solution corresponding to l/r is etkr/r, and there are associated wave-functions - cos k (r — ct), - sin k (r — ct), which are, of course, particular cases of the wave-function in which /(T) is an arbitrary function which is continuous (D, 2). § 6-13. In the case of the conduction of heat the fundamental equation possesses solutions of the form (I) where R, 0, O satisfy the same differential equations as before but T is of type a exp (- Cooling of a Spherical Solid 353 where a is a constant and h2 is the diffusivity. Thus there are solutions of type - cos kr . e~w , - sin kr . e- which depend only on r and t. The second of these is the one suitable for the solution of problems relating to a solid sphere. If, in particular, there is heat generated at a uniform rate in the interior of the sphere the differential equation for the temperature 6 is ot where 6 is a constant. There is now a particular integral — b*r2/6h2 which r)f) must be added to a solution of -^ = h2V29. ct If initially 6 *= 00 throughout the sphere, 60 being a constant, and the boundary r = a is suddenly maintained at temperature 01 from the time t = 0 to a sufficiently great time T, the condition at the surface is satisfied by writing while the initial condition is satisfied by writing* _ As £ -> oo, 0 tends to the value 0t + — - • . 2 -- ; and ^- to the value 62r — ^r-, so that the flow of heat across the surface is, per second, O K Writing 62 = — and h2 = — , where p is the density and a the specific heat of the substance, we have the result that the rate of flow of heat across the surface is 4Q7ra3/3, a result to be anticipated. _ If, on the other hand, the initial temperature is 0t -\ -- 7^2 the surface of the sphere radiates heat to a surrounding medium at temperature 02 at a rate E (9a — 02) per square centimetre, where 6a is the (variable) surface temperature of the surface of the sphere, the solution is e - B - ~ + - XDne~n2M sin nr. on* r * The constant Dm is obtained by Fourier's rule from the expansion of 00 - 6l - ^ 2 (a2 - ra) in a sine series. B 2 354 Po/ar Co-ordinates The surface condition is satisfied by writing R - fl 4- a6* (^ + 2 + where <pm is the with root of the transcendental equation The initial condition gives Fr = £ Dn sin nr, m-l where and the extended form of Fourier's rule gives z- K)' Ea - E (Ea - 2F K2<l>m2 4- (Ea - K}* f« . (r^m\ 3 I~T~W~TF W\ r-sm I ) dr These results have been used by J. H. Awbery* in a discussion of the cooling of apples when in cold storage. § 6-21. Legendre functions. The method of differentiation will now be used to derive new solutions of Laplace's equation from the fundamental i , . 1 , 1 , r + z solutions - and ^- log - . r 2r & r - z After differentiating n times with respect to z the new functions are of form r-n~lQ; consequently we write n i Lt 2 g - ''n^>- n\ dz«\2r^r-z) and we shall adopt these equations as definitions of the functions Pn (p.) and Qn (p) for the case when n is a positive integer and 6 is a real angle. The first equation indicates that there is an expansion of type (^•2 tynfii I /Tf2\ j — — \* — - ~P ( ii\ I n I *?* I Y ^ti-/ fJi i~ i4/i — ^j />«w+l •*• n \r^)i I ^^ I n— 0 and this equation may be used to obtain various expansions for Pn(fi). Thus 1.3 ... (2n- 1) 3»^= 1.2..., V-+...] - ~ = (-)- F - n, n . (7), vol. iv, p. 629 (1927). Hobsorfs Theorem 355 where F (a, 6; c; x) denotes the hypergeometric series 1 + a'bx . M«Ji!L6J*±JQ *;2 , + l.c*4" f.2.c(c + 1) "" § 6-22. Hobsorfs theorem. The first expansion for Pn (^) is a particular case of a general expansion given by E. W. Hobson*. If / (x, y, z) is a homogeneous polynomial of the nth degree in x, y, z, r2V2 r*V4 X 2 (2?) 2.4 When / (x, y, z) = zn this becomes 1 , ^3) '"J ; (*J y' Z)' r n (n - 1) 2 n (» -J)Jn -_2) Jn -3) 4 _ 1 A 1Z 2 (2/1 - : 1) z + 274 (271 - 1) (2n - 3) "'J' which is equivalent to the expansion for n I r~n~lPn (^). Assuming that the theorem is true for / (x, y, z) = zn it is easy to see that the theorem must -also be true for / (x, y, z} = (£x -f r^y + ^)n, where |x + 77^ + ^ is derived from z by a transformation of rectangular axes, for r\ /"\ ^\ ^\ such a transformation transforms ^- into ^ ^ -f -n ^- + t «- and leaves S^ ra ' cy dz V2 unaltered. To prove that the theorem is true in general it is only necessary to show that / (x, y, z) can be expressed in the form / (x9 y,z)= 2 As ($8x + w + £,*)», s=l where the coefficients As are constants. To determine such a relation we choose k points such that they do not all lie on a curve of degree n and such that a curve of degree n can be drawn through the remaining k — I points when any one of the group of k points is omitted. Let £s, rjs, £s be proportional to the homogeneous co-ordinates of the sth point and let i/js (x, y, z) = 0 be the equation of the curve of degree n which passes through the remaining k — 1 points. Assuming that a relation of the desired type exists we operate on both sides of the equation with the operator i/j8 ( ^— , ^— , ^ - j . The result is ' ~dy' Giving s the values 1, 2, ... k all the coefficients are determined. Since a curve of the nth degree can be drawn through \n (n -f- 3) arbitrary points, * Proc. Land. Math. Soc. (1), vol. xxiv, p. 55 (1892-3). 23-2 356 Polar Co-ordinates the number k should be taken to be \ (n + 1) (n + 2), which is exactly the number of terms in the. general homogeneous polynomial / (#, y, z) of degree n. The coefficients A3 could, of course, be obtained by equating coefficients of the different products xaybzc and solving the resulting linear equations, but it is not evident a priori that the determinant of this system of linear equations is different from zero. The foregoing argument shows that with our special choice of the quantities gs , 77 5 , £8 the determinant is indeed different from zero because with a special choice of /, say the equations can be solved. The solution is, moreover, unique because if there were an identical relation 0 = S Cs (tsx + w + £,*)», 3=1 the foregoing argument would give O^nlC^tf,,^, £.). Hobson's theorem has been generalised so as to be applicable to Laplace's equation for a Euclidean space of m dimensions. Writing V . = -£+^+ + ** m -^^ •" [ ' and using/ (^ , a;2, ... xm) to denote a homogeneous polynomial of degree n, the. general relation is \' 4' "' 9 [r'2Vm2 1 ~ 2 (m + 2n - 4) + ^ 2n - 6) § 6-23. Potential functions of degree zero. When n = 0 the differential equation satisfied by the product U = 0O may be written in the form a2 17 _ ~ ' U f ^ f <W 1 -4. ^ where s = - -,— ^— 9 = ~^ — 5 ^ log.tan - . ) I ~ p.2 ] smO &i 2 It follows that there are solutions of type where /is an arbitrary function and/ (u) = This solution may be written in the form Differentiation of Primitive Solutions 357 where F is an arbitrary function. The general solution of Laplace's equation of degree zero may thus be written in the form r where F and G are arbitrary functions*. The general solution of degree — 1 may be obtained from this by inversion and is \ „ (x — ii - 0 >r + zj r \r + z. Solutions of degree — (n 4- 1) may be obtained from the last solution by differentiation. In particular, there is a potential function of type ^ H ix + i ~~ 8z» [r\z + r which is of the form r-"-1^ (0) em*. The function must consequently be expressible in terms of Legendre functions. When m is a positive integer equal to or less than n we have in fact the formula of Hobson )n n /T .L •»?Amn (/ (ITT) J = {~}n (n ~ m) ! r"n"lp» When m is a positive integer greater than or equal to n we have the expansion fl ^ (9-n 1\ 1 ** (9n ^\ 9m 1 , . \ L . O . . . \LTl — 1) L . o . . . I 4 /I — *> l £i 71 — 1 , . : V ) I ~2V~1 / rn^n ~^ ~2n~? m~n I 1 (m ~~ n) + r2n-I7^ rTm-nV2 ~~ j 2 (w - 7l) (m - ?* + 1) + ... ... 4- — _j — (m — n) (m — n+ 1) ... (m — which may be used to define the function x (0) in this case. In particular, we have the relation an_ r ___! dzn [r (z -f r When this is used to transform the expression for r~n-lPn" we find that| * W. F. Donkin, Phil. Trans. (1857). t This formula is given substantially by B. W. Hobson, Proc. London Math. Soc. (1), vol. xxn, p. 442 (1891). Some other expressions for the Legendre functions are given by Hobson in the article on "Spherical Harmonics" in the Encyclopedia Britannica, llth edition. 358 Polar Co-ordinates EXAMPLE Prove that if m is a positive integer m pi ?r J 0 T2 / T J o , elmada z + ix cos a -f ii/ sin a r \z + r \ ( x — z i , . . . v ,• log (z -f ix cos a + it/ sin a) etmada = - - — r ~~ -\- r / § 6-24. Upper and lower bounds for the function Pn (/x). We shall now show that when — 1 < /x < 1 the function Pn (/x) lies between — 1 and -f 1. This may be proved with the aid of the expansion — --" cos n>8 -f- A . -~ - * - — - cos (n — 2) 6 1.3 1.3 ... (2n- 5) + n 2.4-2.4... (2n-4rwv" ' ' ' which is obtained by writing (1 - 2x cos 9 -f x2)~i - (1 - xe*)-* (1 - ze-'T** and expanding each factor in ascending powers of x by the binomial theorem, assuming that | x \ < 1. It should be observed that each coefficient in the expansion is positive, consequently Pn (/*) has its greatest value when 6=0 and /x, = 1, for then each cosine is unity. If, on the other hand, we replace each cosine by — 1, we obtain a quantity which is certainly not greater than Pn (/x). Hence we have the inequality - i < P. (/t> < i, fc* _ 1 < ,. < 1. When n is an odd integer Pn (/x) takes all values between — 1 and + 1, but when n is an even integer Pn (ju) has a minimum value which is not equal to — 1. This minimum value is — | for P2 (p) and — f for P4 (/x). § 6-25. Expressions for the Legendre polynomials as nth derivatives. Lagrange's expansion theorem tells us that if z =-- IJL + acf) (z), the Taylor expansion of/' (z) -v- in powers of a is of type (tjJL az = - - 1 . a2)*, - = (1 - 2/xa + a2)-*; Formulae of Rodrigues and Conivay 359 a comparison of coefficients in this expansion and the expansion (1 - 2/ta-f a2)-* = S a»Pn (M) o gives us the formula of Rodrigues, p« ^ - 2-4l £- [("3 ~ 1)nj- If, on the other hand, we write c£ (z) - 2 (V~z - 0, we have 3 = /z 4- 2a (Vz — t)9 z - 2a Vz + a2 = a2 - 2a^ + ti, = a ± a2 — 2a^ -f M, n + 1 / M = ?" _31 \vV/ n ! 3/xn vV 1 cn (r — t)n Hence or ....- ,-t * » i :. i ~\(rdr)n r This formula is due to A. W. Conway, the previous one to E. Laguerre. Replacing t by z we have the following expression for a zonal harmonic 1 „ , . 1 3n (^-^Y r ' z and r being regarded as independent. §6-26. The associated Legendre functions. The differential equation (II) m of § 6-11 is transformed by the substitution 0 = (1 — p,2)2 P to the form but this equation is satisfied by P = j-~ , where v is a solution of Legendre's equation , //27) fin (I-/**) ^,-2/*;fc+»<»+l)'-0, particular solutions of which are Pn (/x) and Qn (/x). 360 Polar Co-ordinates Hence we adopt as our definitions of the functions Pnm (/z) and Qn™ (/x) for positive integral values of n and m m o nm - 1 < /x< 1. With the aid of these equations we may obtain the difference equations satisfied by Pnm (p.) and Qnm (//,) : (TI - m 4- 1) Pwr?+1 - (2n 4- 1) /zPn™ + (n + m) Pmn^ = 0, +l = 2mp.Pnm - (n + r/i) (n - m + 1) Vl P"*,,., - jzPn™ - (n - m + 1) Vl - Pmn+1 - /xPww 4- (n 4- w 1 - (n 4- m 4- 1) fi.Pnm - (n - m 4- 1) P^+i, and the following expressions for the derivative (I-/*') ^ Pn» (/,) - (n + 1) /JV» (/z) - (^ - m + 1) P-n,! (/x) - (w + w) P-w_i (/x) - n/iPn~ (/x). Similar expressions hold for the derivative of Qnw (/*)• Expressions for the Legendre functions of different order and degree n are easily obtained from the difference equations or from the original definitions. In particular P0°= 1- P!° - cos 0, P^ - sin 6. P2° = i (3 cos2 19-1), Pa1 - 3 sin 9 cos 0, P22 - 3 sin2 5. P3° - \ (5 cos3 0-3 cos 0), P31 - | sin 0 (15 cos2 0 - 3), P32 - 15 sin2 9 cos 9, P33 - 15 sin3 0. P4° - I (35 cos4 0-30 cos2 9 4- 3), P^ - J sin 0 (35 cos3 0-15 cos 0), P42 - £ sin2 0 (105 cos3 0 - 15), P43 - 105 sin3 0 cos 0, P44 - 105 sin4 0. P6o = j (63 cos5 0-70 cos3 04-15 cos 0), Pg1 - I sin 0 (315 cos4 0 - 210 cos2 0 4- 15), P62 - \ sin2 0 (315 cos3 0 - 105 cos 0), P63 ^ | Sin3 ^ (945 Cos2 0 _ 105), P64 - 945 sin4 0 cos 0, P55 - 945 sin5 0. A ssociated Legendre Functions 361 EXAMPLES 1. Prove that if m and n are positive integers 2. Prove that ai ~~ * cf") ^ Pn?W (/i) 6*m^ =• (» + m) (w + w - 1) rn~l Pn (jx + id] [r""n""1 Pr*m (fz) ^^ = "" r~n~2 Pn+lWl (w ~ m + 1J (n ~ § 6-27. Extensions of the formulae of Rodrigues and Conway. By differentiating the formula of Rodrigues m times with respect to ^ we obtain the formula We shall use a similar definition for negative integral values of m and shall write Expanding by Leibnitz's theorem we obtain n— m (n __ w\ f 72, 1 _ V VAf> AA6^ ' __ _ ~ Comparing the two series, we obtain the relation of Rodrigues 362 Polar Co-ordinates This may be derived also from the equations of Schende / \m /I i ,,\ 2 //« = (~> _ (~-^i -— rfu - nn+ 2«(n-m)!\l- p.) d^1^ > ~- (n-m)!\l- p. which may likewise be proved with the aid of Leibnitz's theorem. We have in fact ^—[(/x-ir-^+i)^] nl (n-m)\ (n±m] ! _s " ^ *T(^^ ^-^T^)"! (m + «) ! (/A ' (^+ j ' By differentiating Conway's formula m times with respect to t and m multiplying by (r2 — £2)2 we obtain the formula " ' 0. . - ,, rn+i » yry v ; v ' (n — m)\ \rdrj r Making use of the formula (C) we may also write Changing the sign of m we have This formula also holds for m > 0. § 6-28. Integral relations. The Legendre functions satisfy some interesting integral relations which may be found as follows : Writing down the differential equations satisfied by Pnm (/x) and Ptk (/z) let us first put k = m and multiply these equations respectively by Pf and Pnm and subtract, we then find that + (n - I) (n + I + 1) PnmP^ = 0. Integral Relations 363 Integrating between — 1 and + 1 the first term vanishes on account of the factor 1 — /x2 and so we find that if I ^ n Next, if we put I = n and multiply by Pnfc, Pnm respectively and subtract we find in a similar way that if ra2 ^ k2 To find the values of the integrals in the cases I = n, k = m we may proceed as follows : If we multiply the first difference equation by Pmn+1 (/x) and integrate between — 1 and + 1 we obtain the relation (n - m + 1) f1 [P*'B+1 (M)]2 dp - (2n + 1) P iiPmn+lPnm rf/x, J-i J-i while if we multiply it by Pmn_! (/x) and integrate we obtain the relation (n + m) f1 [P-n_x Oz)]2^ = (2n -f 1) f l /zP^P'V^. J-i J-i Changing ?i into n — 1 in the previous relation we find that (2n + 1) (n - m) [' [Pw« (^)]2 rf^ = (2w - 1) (n + m) f [P-n_t (/x)]2 rf/i. J-i J-i But Therefore [Pnm (/*)? ^= l«-3« ... (2m - 1)* j^l - M«)» dp = ^-j (2m) ! and so f ' [P.* (^ ^ = 9-^rT r ^ ' ]-i r/J r 2n+l(n~m)\ Let us next multiply the difference equations f (1 - M2) —^ = n^P^ -(n- m) Pn- by (1 — ijP)-lPmn^ and (1 — /x2)~1Pww respectively and add. Integrating between — 1 and -i- 1 we obtain the relation (n + m) [P-W]* - - (n - m) = 0 if m > 0. 364 Polar Co-ordinates Now f1 [Pnm (^)]\4r~^ I2.32...(2m- I)2 f1 (I-/*2)" = 2. (2m- 1)!. MIL. r f1 rr» «, / xno ^M- 1 (?l -f m) ! Therefore [Pnm (M)] _ 2 = — . V- — - f . These relations are of great importance in the theory of expansions in series of Legendre functions. See Appendix, Note m. § 6« 29. Properties of the Legendre coefficients. If the function / (x) is integrable in the interval — 1 < x < 1, which we shall denote by the symbol /, the quantities /> 1 (3*} doc (I) are called the Legendre constants. If these constants are known for all the above specified values of n and certain restrictions are laid on the function/(x) this function is determined uniquely by its constants. An important case in which the function is unique is that in which the function (I — x2)%f(x) is" continuous throughout /. To prove this we shall show that if <f) (x) = (I — #2)~i</r (x), where iff (x) is continuous in /, then the equations f1 <f>(z)Pn(x)dx=0 (n= 0,1,2,...) (II) j-i imply that $ (x) = 0. The first step is to deduce from the relations (II) that ( <f>(x)xndx= 0 (TI= 0, 1,2, ...). This step is simple because xn can be represented as a linear com- bination of the polynomials P0 (x), Px (x), ... Pn (x). The theorem to be proved is now very similar to one first proved by Lerch*. The following proof is due to M. H. Stonef. If ^ (x) ^ 0 for a value x = £ in / we may, without loss of generality, assume that $ (£) > 0, and we may determine a neighbourhood of £ throughout which <f> (x) > m > 0. Now if A > 0 the polynomial p (x) *== A — %A (x — £)2(x2 -f 1) is not negative in / and has a single maximum at x == £. We choose the constant A so that in the above-mentioned neighbourhood of' f there are * Acta Math. vol. xxvn (1903). f Annals of Math. vol. xxvii, p. 315 (1926). Theorems of Lerch and Stone 365 two distinct roots of the equation p (x) = 1 which we denote by xl , x2 , the latter root being the greater. We thus have the inequalities 0< p (x) < 1, - 1 < x< xly X2< X < 1, p (x) > 1, <f> (x) > m, xl < x < x2y ifj (x) > - M , - Kx< I, w where M is a positive quantity such that — M is a lower bound for the continuous function 0 (x). Writing pn (x) — [p (#)]n, we have <t>(x)pn(x)dx=0, n=l,2. (A) L On the other hand fx, rx, <£ (*0 Pn (x) dx> m\ pn (x) dx, Jxi Jxl f ' <f>(x)pn(x)dx> - M\ l (1 -x*)-ldx, J-i J-i r1 f1 <f> (x) pn (x) dx > - M\ (I- a;2)-* dx, J x, J xt r1 (x* * r1 ^ (^) Pn (x) dx> m \ pn (x) dx — M \ (1 — #2 J-l JXi J-l f*1 > m pn (a?) rfa; — rrM. fx, Since pn (x) dx -> oo as n -> oo we can choose a number N such that the right-hand side is positive for n > N. This contradicts (A) and so we must conclude that <f> (x) = 0 throughout /. Lerch's theorem is that if ijj (x) is a real continuous function and /•i xn 0 (x) dx = 0 for tt = 0, 1, 2, ... to oo, then iff (x) = 0. Jo By Weierstrass's theorem the function i/j (x) may be approximated uniformly throughout the interval (0, 1) by a polynomial G (x). In other words, a polynomial G (x) can be chosen so that ifj (x) = G (x) -f 86 (x), where | 0 (x) \ < • 1, and 8 is any small positive number chosen in advance. Now if t/j (x) is not zero throughout the interval (0, 1) we can choose our number 8 so that Ji ri o Jo But, since G (x) is a polynomial, we have 366 Polar Co-ordinates Therefore f V (*) [<A (*) ~ 8 Jo or P[<A (a)]2 da = s[ 9 (x) 0 (a) J o Jo f1 Jo [ o >o This contradicts (B) and so we must have 0 (x) = 0. Putting x = e~* roo we deduce that if er* $ (t) dt = 0 f or 2 > 0, and </> (£) is continuous for Jo t > 0, then f/> (J) = 0. EXAMPLES 1. When m and n have positive real paits A Qm (z) Qn (2) dz - *(n -hi)- *(m -h 1) -2l7r [(« - T) sin (\m -f- Jw) where ^(z) = log r (2), A = (m - rt) (m -f n-f I) H = /*(}w + J, J'" + 1) j ;,r [S. (* Dharand \ (1 Shabde, Hull Calcutta Math Soc. v. 24, 177-186 (1932). also that v\ith the same notation Qm (z) Qn (2) dz - t (m + 1) - * (w -f 1) ((^•ne.sh Prasad, Proc Henarex Math. Soc. v 12, pp. 33-42, 19.) 2. Show by means of the relation that when n is a positive integer the equation Pn (ft) = 0 has n distinct roots which all lie in the interval — 1< /x< 1. 3. Prove that when m and n are positive integers 2,m+n-\i <(m . W\u4 M4-2^m*-nP ^^P (2^^ - .i7^^71;-!. i + 2) rrn(.) rw (zj a~ - {mlnl}2 (2m + 2n 4_ X) ,. [E. C. Titchmarsh. An elementary proof of this formula is given by R. 0. Cooke, Proc. London Math. Soc. (2), vol. xxin (1925); Records of Proceedings, p. xix. Green's Function for a Sphere 367 § 6-31. Potential function with assigned values on a spherical surface S. Let P, P' be two inverse points with respect to a sphere of radius a. If 0 is the centre of the sphere we have then OP. OP' -a2, and 0, P, P' lie on a line. The point 0 is sometimes called the centre of inversion. If P lies inside the sphere, P' lies outside ; if P is outside the sphere, P' is inside. If P is on the sphere, P' coin- cides with P. If P describes a curve or surface P' will describe the inverse curve or surface and it is clear that a curve or surface will intersect the sphere at points where it meets its inverse. If a curve or surface inverts into itself it must intersect the sphere S orthogonally at the points where it meets it because at these points two coi sccutive inverse points lie on the surface and on a line through 0. This line is then a tangent to the surface and a normal to 8 at the game point. If Ms is any point on S the triangles 0PM s, OMSP' are similar, and we have OP PM8~P'M,' If charges proportional to OP and — OMS are placed at P and P1 respectively, the sum of their potentials at any point Ms on S will be zero. Writing OP =_r, PM =-- R, P'M - R' ', where I/ is any point, we see that the function __ I a 1 ^PM~P"r*^ is zero when M is on S and is infinite like -„ at the point P. We shall call Zi this function the Green's function for the sphere. GP}J is easily seen to be a symmetric function of the co-ordinates of P and M, £ r if JT is the inverse of Jf we have Oif OP P'M ^ PM7' The point P' is called the electrical image of P and (7/>3/ represents the potential at M when the sphere $, regarded as a conducting surface at zero potential, is influenced by a unit charge at P. When a becomes infinite and 0 recedes to infinity the sphere becomes a plane, P' is then the optical image of P in this plane, and the virtual charge at P' is equal and opposite to that at P. 368 Polar Co-ordinates Now let OM = r' and POM = <*>, then 2 __ 2rr cos co, __ 2r' cos co, a a (ai + r2 - 2ar cos a>)' Let (r, 0, <£), (r, 0', </>') be the spherical polar co-ordinates of the point P and a point Ms on the surface of S, then the theorem of § 2-32 tells us that if a potential function V is known to have the value F (#', <£') at a point M8 on $ then an expression for V suitable for the space outside S is o (a2 -f r2 - 2ar cos co) while a corresponding expression suitable for the space inside S is* 1 fir r2n a (a2 - r*)F (#',</>') sin 6' V (r, 9, (/>) - -i- rffl' <Z<A' - ~— 7 ~~7 1^ ^T" - ^ ' ^; 47rJo Jo (a2 + r2 - 2ar cos w)* When the sphere becomes a plane the corresponding expression is , , ± /(«',»') the upper or lower sign being taken according as z ^ 0. In this case / (x1 ', y') = V (xf ', ?/', 0) is the value of F on the plane 2=0. § 6' 32. Derivation of Poissons formula from Gauss's mean value theorem. Poisson's formula may also be obtained by inversion, using the method of Bocher. Let us take P' as centre of inversion and invert the sphere 8 into itself. The radius of inversion is then c = (r02 — a2)* = - (a2 — r2)i, where c is the length of the tangent from P' to the sphere, it is real when P' is outside the sphere and imaginary when P' is within the sphere. (In Fig. 28 OP'-r0.) Let Q, Q' be two corresponding points on $, then the relation between corresponding elements of area is cr \4 Writing dS' = a*d£l', dS = a2dfl, where d&' and dO are elementary- solid angles, we have dtt = ( Cp^) d£l = (a2 - r2)2 (r2 + a2 - 2ar cos w)-2 dQ, \U- . x v^/ where co is the angle between OQ and OP. * This is generally called "Poissoi^s integral," both formulae having been proved by S.D. Poisson, Journ. £cole Polyt. vol. xix (1823). The formula for the interior of the sphere had, how- ever, been given previously by J. L. Lagrange, ibid. vol. xv (1809). Poisson's Formula and its Generalisations 369 Now if V1 'Q> is a potential function when expressed in terms of the co-ordinates of Q', the function P'Q Q' is a potential function when expressed in terms of the co-ordinates of Q, consequently the mean value theorem glV6S 47rF0' = a (a2 - r2)2 f F0dQ [r2 + a2 - 2ar cos «]*, CI* J and since c. F0' = P'P. VP, we have crV0' = (a2 — r2) FP, and our formula is the same as that derived from the theory of the Green's function. This method is easily extended to the case of hyperspheres in a space of n dimensions. The relation between the contents of corresponding elements of the hyperspheres is now *?_' _ f?'^\n~l - (JLVn~2 - ( cr Vw~2 dS-\P'Q) ~(P'QJ ~~\a.PQ> ' while the relation between corresponding potentials is Writing the mean value theorem in the form the generalised formula of Poisson is cr n *«2\n-2 f /rr\n f — jr - ) FP US' = ( - J F0 dS [r2 + a2 - 2ar cos a>] 2 / J VQ- / J 71 or VP J AS' = ± a"-2 (a2 - r2) J FQ cZS [r2 + a2 - 2ar cos cu] " ^. § 6*33. /Some applications of Gauss's mean value theorem. The mean value theorem may be used to obtain some interesting properties of potential functions. In the first place, if a function F is harmonic in a region .R it can have neither a maximum nor a minimum in R. If the contrary were true and F did have a maximum or minimum value at a point P of R the mean value of F over a small sphere with centre at P would not be equal to the value at P. If now the sphere is made so small that it lies entirely within jR, Gauss's theorem may be applied B 24 370 Polar Co-ordinates and we arrive at a contradiction. Since a function which is continuous over a region consisting of a closed set of points has finite upper and lower bounds which are actually attained, we have the theorem : // a Junction is harmonic in a region R with boundary B and is con- tinuous in the domain R -|- B, the greatest and least values of V in the domain R |- n are attained on the boundary B. One immediate consequence of the last theorem is that if the function V is harmonic in R, continuous in R -f B and constant on B it is constant on R -f B. This theorem is important in electrostatics because it tells us that the potential is constant throughout the interior of a closed hollow conductor if it is known to be constant on the interior surface of the conductor. Another interesting consequence of the theorem is that if the function V is harmonic in R, continuous in R \ B and positive on B it is positive in R 4- B. For if it were zero or negative at some point of R the least value of V in R would not be attained on the boundary*. This theorem may be restated as follows: If \\ and V2 be functions harmonic in R and continuous in R 4- B, and if Vv is greater than (equal to or less than) V2 at every point of B, then \\ is greater than (equal to or less than) V2 at every point of E -f B. A converse of Gauss's theorem, due to Koebe, is given in Kellogg's Foundations of Potential Theory, p. 224. § 6' 34. The expansion of a potential function in a series of spherical harmonics. If V (x, y, z) is a potential function which is continuous throughout the interior of a sphere S and on its boundary, and whose first derivatives ^ , ^ , ^ arc likewise continuous and the second ex en cz derivatives finite and integrable (for simplicity we shall suppose them to be continuous) then V admits of a representation by means of Poisson's formula and it will be shown that V can be expanded in a convergent power series in the co-ordinates x, y, z relative to the centre of S. Writing cos o» — cos 6 cos 6' -f sin 8 sin 6' cos (</> — <£') — //,, we have 2 /2\ oo n l P. (M) | r | > a, (a2 4- r2 - 2 ., - „ - (a1 -f r1 -- 2<7rcosoo)* n Substituting in the expressions for V we may integrate term by terrr. because the series are absolutely and uniformly convergent on account ot the inequality | Pn (p) \ < 1. * See a paper by G. E. Ray nor, Annals of Math. (2), vol. xxm, p. 183 (1923). Expansion of a Potential Function 371 We thus obtain the expansions n=0 " 7- S ( n=0 where in each case Sn (6, ft = I \' f 'V (0', 47Tj() JO sn The function rnSn (0, <f>) is called a,spherical harmonic or solid harmonic of degree n, it is a polynomial of the nth degree in x, ?/, z arid is a solution of Laplace's equation because rnPn (/*) is a solution. The function Sn (9, <f>) is called a surface harmonic, it may be expressed in terms of elementary products of type Pnm (cos 0) e*tn* by expanding Pn (fji) in a Fourier series of type Pn(/A) = 2 Fnm(0,0')e"» <+-*'>. in— —n By expanding rnPn (p) in a series of this form and substituting in Laplace's equation (in polar co-ordinates) we get a series of typeSCfwe*m(*""*/) each term of which must be separately zero, consequently each term in our expansion of rnPn (p,) is a solution of Laplace's equation and is a poly- nomial of degree n in x, y and z. Similarly, if r', 0' , c/>' are regarded as polar co-ordinates of a point (x'9 yr, z'), r'HPn (IJL) is a solution of Laplace's equation relative to the co-ordinates of this point. We infer then that Fn™ (0, 8') - AnmPnm (cos 0) Pn-m (cos 0'), where Anm is a constant to be determined. We thus have the result that Sn(0,<j>)= S £nmP«m (cos 6) e"»*, m=*—n where Bn™ - .— ^nm f f ^ F (0f, <f>') Pn~m\cos 9') er™*' sin 0'd9'dcf>'. 477 J 0 J 0 • To determine the constant Anm we consider the particular case when V = rnPMm (cos 9) eim*, F (9', </>') = anPnm (cos 0') e*™+\ then (2i/+l) S, (fl, <£) - anPw^ (cos (9) e'™* i/ - n = 0 ^=^w, and consequently ,2™ 4- 1 X- -4^ or ^4nw — (— )m. 24-2 Anm ^ f "" Pnm (cos 0') Pn-« (cos 9') sin 0'd9'd<f>' Jo Jo 372 Polar Co-ordinates Hence we have the expansion Pn (p) - S (-)mPnm (cos 6) Pn~™ (cos 0') e"»<*-*'>. m**—n § 6*35. Legendre's expansion. Transforming the last equation with the aid of the relation of Rodrigues, U ~~ m -m /..\ __ - we obtain Legendre's expansion* 00 (rvi _ . Yf) \ t Pn (p) = 2 S — - - j Pnm (cos 6) Pn» (cos 0') cos m (<£ - f ) m—l \'^ ' '"7 i + Pn (cos 0) Pn (cos 0'), and the expression for Bnm may be written in the alternative form 1 (rt — f f w 1 10 One simple deduction from the expansion for Sn (6, <f>) is that a simple expression can be obtained for the mean value of Sn (9, </>) round a circle on the sphere. Let the circle in fact be 0 = a, then the mean value in question is obtained by integrating our series for Sn (9, </>) between 0=0 and $ = 2-n- and afterwards dividing by 2ir. The result is that tin (0, </>) = Bn»Pn (cos a). Now when 0=0, Pnm (cos 9) = 0 except when m = 0, and then the value is unity, hence Sn (0, 0) = ,Bno, and so Sn (6, <f>) = flfn (0, 0) Pn (cos «), where the coefficient Sn (0, </>) is the value of Sn (9, <f>) at the pole of the circle. This theorem may be extended so as to give the mean value of a function / (9, </>), which can be expanded in a series of type The result is ni=0 n=0 If the analytical form of the function /is not given, but various graphs are available, the present result may sometimes be used to find the coefficients in the expansion /(0,<£)= 2 C'nPn(cos0)+ S E Pwm(cosfl)[-4nmcosm^ + 5nmsinm^]. n=0 n==l m=0 To use the method in practice it is convenient to have a series of curves in which / is plotted against <f> for different values of 0 and a series of curves in which / is plotted against 9 for different values of <f>. The two * Legendre, Hist. Acad. Sci. Paris, t. n, p. 432 (1789). Expansion of a Polynomial 373 meridians <f> = /3 and <f> = /? 4- TT may be regarded as one great circle with Mean values round "parallels of latitude" for which 6 has various constant values will give linear equations involving only the coefficients Cn . Since Pn (0) = 0 when n is odd and Pnm (0) = 0 when n is even and m is odd, the mean values round meridian circles will give equations involving only the coefficients Anm and Bnm in which both m and n are even, but terms of type Cn will also occur. To illustrate the method we shall suppose that the function / (9, <f>) is of such a nature that spherical harmonics of odd order or degree do not occur in the expansion and that a good approximation to the function may be obtained by taking terms of orders and degrees up to n = 4 and m — 4. We have then to determine the nine coefficients <70, C2, 64, A22, £22, ^444, J544, ^442, J342. Three of these may be determined from the mean values of / round parallels of latitude, say 0 = ^, 0 = =, 0 = -. Two equations connecting A22, A^, A42 may be 2 o u obtained from the mean values of / round the meridians </> = 0, TT and <f> = ~, -0- , while two equations involving jB22, JS42, A£ may be obtained £ 2i . from the mean values round the circles <A = ~ , -.- : J> = —r , c6 = — - . ^ 4 4 r 4 T 4 Further equations may be obtained from the mean values round the circles ,, 77 ITT , 77 4?r , 2?r STT , 5?r HTT ^ = 6' T;<^ = 3' "3;*="3"' '3"^^ 6 J ~6~- Having found CQ, (72, C4 from the first three equations and having expressed A22, A^, JS22, 542 in terms of Af with the aid of the next four, two of the last set of equations can be transformed into equations for Af and Bf. When the two sets of curves have been drawn the mean values of / round the different circles may be found with the aid of a planimeter. § 6*36. Expansion of a polynomial in a series of surface harmonics, When rnF (9, <f>) is a polynomial of the nth degree in x, y, z the expansion of F (0, $) in a series of surface harmonics may be obtained in an elementary wav by using the operator V2. Let us write rnF (9, <f>) = fn (x, y, z). The first step is to determine a polynomial /n_2 (x, y, z) such that fn (X> y, Z) - ?*2/n_.2 (X, y, Z) is a solution of Laplace's equation of type rnSn (9, <f>). The equation V2[/n-r*/n_2]=0 gives just enough equations to determine the coefficients in/n_2. To show that the determinant of this system of linear equations does not vanish we must show that y2 rrzf ] =t o 374 Polar Co-ordinates If, however, r2/n_2 were a spherical harmonic of degree n we should have (^2/n_2)/w-2^ — 0 when integrated over the spherical surface, because /n_2 can be expressed in terms of surface harmonics 8m (9, </>) of degree less than n. But this equation is impossible unless /n_2 vanishes identically. Having found /n_2 we repeat the process with /n_2 in place of fn and so on. We thus obtain a series of equations f __ r2f __ r2.Qf J n ' J n—2 ' ^n J f — T2f = T2S from which we find that When n = 2m, where w is an integer and/n = /, the spherical harmonics are determined by the system f)f equations* V2"-/- (2m, 2) (2m 4- 1,3)S0, V2w-2/ = (2m, 4) (2m 4- 1, 5) r2#0 -I- (2m - 2, 2) (2m 4- 3, 7) r2S2, V2m-4y = (2m, 6) (2m 4-1,7) r4/S0 4- (2m - 2, 4) (2m -f 3, 9) r*82 4- (2m - 4, 2) (2m 4 5, 11) r4/SY4, where (a, 6) = a (a — 2) (a — 4) ... 6. Solving these linear equations we find thatf r'^Sw (2m - 2k, 2) (2m 4- 2k 4- 1, 4fc 4- 3) r4 + 2.4~(4]fc -" where 02fc (r, y, 2) = V2m-2kf (x, y, z). The equivalence of the two expressions for S is a consequence of Hobson's theorem (§ 6-22). There is a corresponding theorem for a space of n dimensions. The fundamental formula for the effect of the operator is Vw2 (r2%) - 2p (2p -f- 2q 4- n - 2) r2*>~% -f r*»Vn*vQ, where r2 = o:^ 4- x22 4- ... xn2, t;v = VQ (xl9 x2, ... xn). * If VQ (x, y, z) is a rational integral homogeneous function of degree m, we have V2 (r*%) = 2p (2p + 2q + 1) r2?-2^ -f r2P V2v0. Hence if V2vQ = 0 the effect of successive operations with V2 is easily determined. f G. Prasad, Math. Ann. vol. LXXII, p. 435 (1912). Legendre Functions of i cot 6 375 The equations are now Vn**/= (2m, 2) (2m + n - 2, n) S0, Vn2™-2/ - (2m, 4) (2m -f 7i - 2, 7i -f 2) r2S0 4- (2m - 2, 2) (2m + n, n -f 4) r2S2 , r2*^ (2m - 2i, 2) (2m -f- 2fc + n - 2, 4fc + n) r2\7 2 r4V 4 1 * 4_ _ _ n — V2m-2fc/' 2(4fc + n- 4) ' 2.4(4fc + ra- 4) (4i + rc- 6) '"J ' __ r4*+*-2 _ / a a a\ 2_n - (ifc - 4 + w, n - 2) fe Va^' 8^2 ' '" dxj T ' where 02fc (^ , o:2 , . . . xn) - Vn2w|-2*/ (^ , a?2 , . . . a:n), and / is a homogeneous polynomial of degree 2m. § 6-41. Legendre functions and associated functions. It should be observed that Laplace's equation possesses solutions of type rnPnm (/A) e*tm*, r»Qnm (p.) e±tm*, when n and m are any numbers. It is useful, therefore, to have definitions of the functions Pnm (/x) and Qnm (/z) which will be applicable in such cases and also when fi is not restricted to the real interval — 1 < /* < 1 . The need for such definitions will appear later, but one reason why they are needed may be mentioned here. In an attempt to generalise the method of inversion for transforming solutions of Laplace's equation* it was found that if Y - Jl^-?2 7 - - t r* + a~ Z = - °*- (I) 2(x-M/)' 2(x-iy)' x-iy' (> an(J if / (X, Y, Z) is a solution of Laplace's equation in the co-ordinates X, Y, Z, then (x-iy)-lf(X, Y,Z) is a solution of Laplace's equation in the variables x,y,z. Introducing polar co-ordinates, we find that R = iae1*, r = iae1*, sin 0 = cosec 9. The standard simple solutions of Laplace's equation give rise, then, to new simple solutions of type (sin 0)~* Pnm (i cot 9) ct(n+i>* r~l+m, and we are led to infer the existence of reciprocal relations between associated Legendre functions with real and imaginary arguments and of more general relations when the arguments are complex quantities or real quantities not restricted to the interval — 1 < z < 1 . Definitions of the associated Legendre functions Qnm (z) for all values * Proc. London Math. Soc. (2), vol. vii, p. 70 (1908). 376 Polar Co-ordinates of n, m and z have been given by E. W. Hobson* and by E. W. Barnesf. The definitions adopted by Barnes are as follows : Let z = x -h iy, w = log - — , 25 — 1 » 2»r(l-ra-.s) jmu, e then, if | arg (z — 1) | < TT, ^n™ (^) == — sin 7&7T y (m, n, 5) (2 — I)8 d#, where the integral is taken along a path parallel to the imaginary axis with loops if necessary to ensure that positive sequences of poles of the integrand lie to the right of the contour, and negative sequences to the left. Also Qnm (z) = e~m where Im = TT cosec n7r.Pnm (z), and the upper or lower sign is taken in the exponential factor multiplying Im according as y ^ 0. The functions Pnm (z), Qnm (z) are not generally one-valued. To render their values unique a barrier is introduced from — oo to 1. When'w is not a positive integer and z is not on the cross-cut, Pnm (z) is expressible in the form Pnm (^ - p -~_ — -} AmwF{ - n, n + 1; 1 - m; J (1 - «)}, where F (a, b ; c ; x) denotes the hypergeometric function or its analytical continuation. This formula, which gives a convergent series when | 1 — z | < 2, shows that z = 1 is a singular point in the neighbourhood of which Pnm (z) has the form (,_ 1)-** {(70 + ^(2- 1)+...}. Under like conditions — 2Qnm (z) . sin rnr. F (— m — ri) * -f 1 where, as before, ew = - - . z — 1 The definition of Qnm (z) given by Barnes differs from that given by Hobson, the relation between the two definitions being given by the formula sin HTT [Qnm (z)]B = e~im" sin (n + m) 7r.[Qnm (z)]H. * Phil. Trans. A, vol. CLXXXVII, p. 443 (1896). 4- n».**f IM.~*» ™%i WVTV ^ OT /icmo\ Relations between the Functions 377 It follows from the definitions that PB™ (- z) = e*»-* P.- (z) - 2 5£^T £„« (2), #nm(-2) = -<?„"> (Z).e±"", P"-,-! (Z) = Pnm (2), £"•_„_! (Z) = <?„»» (2) - 7T COt Tfor.P," (Z), Vm <*) ."1 ( sin g»-m (z) r (m - n) = &•» (z) r (- m - »). When m = 0, or when m is an integer, Pnm (z) has no singularity at z=l. This is evident from the expression for Pnm (z) in terms of the hypergeometric function in the cases when m is negative or zero and may be derived from the formula - rl~ m + n) n -n (i+ -_ in the case when m is positive. We add some theorems without proofs. 1°. The nature of the singularity of Pnm (z) at z = — 1 may be in- ferred from the formula + e~m r (- nmr+ n) F{1 ~ m + n' ~ m ~ n> l ~ where 0 = | (1 -f z). When m is a positive integer, x F{m — n, m + n+ l;m+ 1; J (1 — «)} ....... (A) 2°. When in addition n is an integer there are three cases: (1) 0 < n< m. In this case Pnm (z) = 0 but r (1 + n — m) Pnw (2) is a solution of the differential equation. (2) n> m. In this case the formula (A) is valid. (3) n < 0. In this case, if — n > m, p m (z) _ fc2 _ Dim T (m - 71) __ ^n W-iz i) 2T(-m-n)r(m+l) x ^{w — n, ra-fw+ 1 ; m + 1; £(1 — z)}. 3°. If - n < m, Pnm (2) = 0, but F (- m - n) Pnm (z) is a solution of the differential equation. 378 Polar Co-ordinates 4°. When m = 0 and n is not an integer and is not zero, = s rji ~ nll (?L±J _+J) 0< x {log 0 - 20 (1 4- 0 4- ^ (* - ft) + ^ (ft + 1 -f OK where 0 = i (1 4- 3) and 0 (?/) = --.- log F (%)• 2 V / V \ / fa fe \ / Hence Pn(z) has a logarithmic singularity at x — — 1, at which it becomes infinite like TT~I sin UTT . F { — n, n -f 1 ; 1 ; 0} log 0 -f a power series in 0. 5°. When m — 0, we have seen that Pn (z) has a cross-cut from — oo to — 1 ; when, however, \m is not an integer and not zero, Pnm (z) has a cross-cut from — oo to 1, and is therefore not defined by the preceding formulae when — 1 < 3 < 1. It is convenient to have a single value of the function in this interval, and one which is real when m and n are real. It is therefore assumed that as e -> 0 and — 1 < x < 1, Pnm (x) -~ lim e*mnl Pnm (x -f €i) - lim e-^mtpnm (x _ €i) 1 /I -h x\ - 7/ ~~ ,- m; - n™ (x) - I lim {Qn>* (x + ei) + Qn™ (x - €i)} = _ _ - ^ 1^.. . [0Cm> (X) + 2 sin H,TT I (— m - n)1 ' where 6°. The function Qnm (x) has a cross-cut between — 1 and 1. For values of z for which | z \ > 1 the function can be expanded in a convergent power series in 1/3. If | arg (z ± 1) | < TT and (2- - l)irn - (z - l)*m (2 + l)im, / ,n /x sin> + w) ^ r («• + m 4 1) T (J) (2a - l)Jw x F(Jn-f Jm -f 1, jn-f |m + J;w + f ;«-2). The values for cases in which | 3 | < 1 may be deduced by analytical continuation of the hypergeometric function and use of the foregoing definition when z is real. Reciprocal Relations 379 § 6*42. Reciprocal relations*. Barnes has shown that the power series in l/z can, under the foregoing conditions, be expressed in the form O ™(z\ ~ C Vn \Z) ~ O -- ; T-— , i — z j sin (n -f m) TT T (n + m + 1) T (4) ° "" '"sin nit " 2»" r (n -f |) ' Putting z = i cot (9, we have (>nm (i cot 0) - Ci-n-1 sin**1 0 x ^"{| (n + m + 1), \ (n — 7>i + l);n -f |; sin20}. Now F {J (n -f m 4- 1), I (n - m 4- 1) ; n 4- f ; sin2 9} = F {n -f m + 1, n - m + 1 ; n + jj ; sin2 £0} - (cos ifl)-2"-1 .F [m +J,J-w;n-f|; sin2 10]. Therefore Qn« (i cot 0) - CT11-1 2" f* (sin 0)i (cot |0)~W"J x JP [m + A, i - m ; n + | ; sin J0]. But 3- (cot |0) n 4 F {m -f £, J — M; n -f- f ; sin2 £0}. Therefore Qnm (i cot 0) = . ^7 — — (1 sin 0)* P"n-* (cos 0). sin n-n . 1 (— m — n) * -m-t Writing — m — | in place of ft and — n — \ , in place of m, the formula becomes Q~"~\ (i COt 0) -^4~ — Tx (1 sin 0)* PnW (COS 0). ^_m_i v / cos m7T Y (m -f- n -f 1) Again, Barnes has shown that when | 1 — z2 | > 1, -f C2 (2* - l)n^ i (m - 7i), £ (- m - n); - n; I 1 where 2—(--) 2Tj(n+ |) . * Judging from a conversation with Dr Barnes in 1908 he had at that time noted at least one explicit reciprocal relation between the functions Pnm(z) and Qnm(z). 380 Polar Co-ordinates consequently, using again the transformations of the hypergeometric series, we find that P™ (i cot 6) = (2* cosec 0)~ x .F{£-hra, £ — m; n -f- f ; sin2 £0} + rlrT^1!) 6l""(cot ^)n+* -** tt~ ^J + ^;i-^;sin2^} = - sin (w + I) TT. r (w + w + 1) (27r cosec 0)~* lim Ql*l\ (cos 0 - fe), 7T e->0 and so P"n"* (i cot 0) = - - sin UTT . T (- m - n) (2n cosec 0)-* lim Qn™ (cos 0 - t€) ~~w"~* 77 e^Q = - «,- ------ r — v--—'/ --- -— x- (27r cosec 0)~* lim ^nm (cos 0 - fc). r(l 4- m -f n) sm(m + n)ir *->o This is very similar to the reciprocal formula obtained by F. J. W. Whipple*, which may be written in the form n) ^ nm (cosh a) - »___m n ^ " v ; * -m~* EXAMPLES 1. Prove that when n is a positive integer {(/t2 - 1)n log * (1 + ")} - Pn [A. E. JoUiffe, Mess, of Math. vol. XLIX, p. 125 (1919).] 2. Prove that if 2x = <i + t~i, i i, 1, 1 - I) = t* (2 log 2 - i log t) f (|, i; 1; 0 I)' Prove also that - 1 [(4 log 2 - 4 - log 0 ^ (?, i; 1; 0 - 4 {(f)2 (J - J) * + i [(f )2 ~ (S)2] (J + * - i - i) t* + ...}]. [H. V. Lowry, Phil. Mag. (7), vol. n, p. 1184 (1926).] * Proc. London Math. Soc. (2), vol. xvi, p. 301 (1917). Potentials of Degree n + \ 381 3. If K is the quarter period of elliptic functions with modulus k and complementary modulus k* = (1 - &a)i, prove that V " K = ^ W ,P- 2p*"ni-jfc« (1*-*) P (** i; * ; (T+l?) The last series is recommended by Lowry for the calculation of jfiT when k is nearly 1. 4. Prove that if n is a positive integer the equation Pn-f (z) = 0 has no root which lies in the range 1 < z < 3. 5. Prove that if 2n + 1 4= 0 6. Prove that if n is a positive integer Pn(